(* The characters of ℤ/2 are required to write down the Fourier-Stieltjes representation of a binary tensor. *)
Char[L_, A_] := (-1)^(Length[Intersection[L,A]]);
AtomDistr[A_] := 1 + σ12 γ^x12 Char[{1,2}, A] + σ13 γ^x13 Char[{1,3}, A] + σ14 γ^x14 Char[{1,4}, A] + σ23 γ^x23 Char[{2,3}, A] + σ24 γ^x24 Char[{2,4}, A] + σ34 γ^x34 Char[{3,4}, A] + δ123 Char[{1,2,3}, A] + δ124 Char[{1,2,4}, A] + ϵ Char[{1,2,3,4}, A];
(* See Fero's paper ("On patterns of conditional independences and covariance signs among binary variables") for the parameters and construction. We choose a particularly simple, integer solution to the solvability problem for (Subscript[ℒ, 2]) = { [X _ Y], [X _ Z | U], [Y _ U | Z] }. The chosen exponents x are as small as possible which makes the range in which they define a non-negative tensor as large as possible. *)
params = {
σ12 -> 0, σ13 -> σ14 σ34, σ24 -> σ23 σ34, σ14 -> 1, σ23 -> 1, σ34 -> 1,
x12 -> 1, x13 -> x14 + x34, x24 -> x23 + x34, x14 -> 1, x23 -> 1, x34 -> 1
};
(* The distribution P corresponding to these parameters *)
P = Map[#1[[1]] -> 1/16 AtomDistr[#1[[2]]] &, { p0000 -> {}, p0001 -> {4}, p0010 -> {3}, p0011 -> {3,4}, p0100 -> {2}, p0101 -> {2,4}, p0110 -> {2,3}, p0111 -> {2,3,4}, p1000 -> {1}, p1001 -> {1,4}, p1010 -> {1,3}, p1011 -> {1,3,4}, p1100 -> {1,2}, p1101 -> {1,2,4}, p1110 -> {1,2,3}, p1111 -> {1,2,3,4}}] //. params

(* All these tensors indeed satisfy (Subscript[ℒ, 2]). *)
S =
(* [X _ Z | U] *)
p0011*p1001+p0111*p1001-p0001*p1011-p0101*p1011+p0011*p1101+p0111*p1101-p0001*p1111-p0101*p1111==0&&p0010*p1000+p0110*p1000-p0000*p1010-p0100*p1010+p0010*p1100+p0110*p1100-p0000*p1110-p0100*p1110==0 &&
(* [Y _ U | Z] *)
p0011*p0110-p0010*p0111-p0111*p1010+p0110*p1011+p0011*p1110+p1011*p1110-p0010*p1111-p1010*p1111==0&&p0001*p0100-p0000*p0101-p0101*p1000+p0100*p1001+p0001*p1100+p1001*p1100-p0000*p1101-p1000*p1101==0 &&
(* [X _ Y] *)
p0100*p1000+p0101*p1000+p0110*p1000+p0111*p1000+p0100*p1001+p0101*p1001+p0110*p1001+p0111*p1001+p0100*p1010+p0101*p1010+p0110*p1010+p0111*p1010+p0100*p1011+p0101*p1011+p0110*p1011+p0111*p1011-p0000*p1100-p0001*p1100-p0010*p1100-p0011*p1100-p0000*p1101-p0001*p1101-p0010*p1101-p0011*p1101-p0000*p1110-p0001*p1110-p0010*p1110-p0011*p1110-p0000*p1111-p0001*p1111-p0010*p1111-p0011*p1111==0;
(* The bounds from the paper guaranteeing positivity of the tensor. *)
R = 1/3 4^-2 > γ > 0 && 1/3 > ϵ^4 (ϵ+1)^4 && 1/3 4^-3 > δ123 > 0 && 1/3 4^-3 > δ124 > 0; (* infinitesimal *)
Simplify[S //. P]

(* Entropies and score. All summands contribute to the entropy because the parametrized tensors are strictly positive. *)
EntropyExpr[P_] := -(Plus @@ Map[#1 Log[#1] &, P]);
hX = EntropyExpr[{p0000 + p0001 + p0010 + p0011 + p0100 + p0101 + p0110 + p0111, p1000 + p1001 + p1010 + p1011 + p1100 + p1101 + p1110 + p1111}];
hY = EntropyExpr[{p0000 + p0001 + p0010 + p0011 + p1000 + p1001 + p1010 + p1011, p0100 + p0101 + p0110 + p0111 + p1100 + p1101 + p1110 + p1111}];
hZ = EntropyExpr[{p0000 + p0001 + p0100 + p0101 + p1000 + p1001 + p1100 + p1101, p0010 + p0011 + p0110 + p0111 + p1010 + p1011 + p1110 + p1111}];
hU = EntropyExpr[{p0000 + p0010 + p0100 + p0110 + p1000 + p1010 + p1100 + p1110, p0001 + p0011 + p0101 + p0111 + p1001 + p1011 + p1101 + p1111}];
hXY = EntropyExpr[{p0000 + p0001 + p0010 + p0011, p0100 + p0101 + p0110 + p0111, p1000 + p1001 + p1010 + p1011, p1100 + p1101 + p1110 + p1111}];
hXZ = EntropyExpr[{p0000 + p0001 + p0100 + p0101, p0010 + p0011 + p0110 + p0111, p1000 + p1001 + p1100 + p1101, p1010 + p1011 + p1110 + p1111}];
hXU = EntropyExpr[{p0000 + p0010 + p0100 + p0110, p0001 + p0011 + p0101 + p0111, p1000 + p1010 + p1100 + p1110, p1001 + p1011 + p1101 + p1111}];
hYZ = EntropyExpr[{p0000 + p0001 + p1000 + p1001, p0010 + p0011 + p1010 + p1011, p0100 + p0101 + p1100 + p1101, p0110 + p0111 + p1110 + p1111}];
hYU = EntropyExpr[{p0000 + p0010 + p1000 + p1010, p0001 + p0011 + p1001 + p1011, p0100 + p0110 + p1100 + p1110, p0101 + p0111 + p1101 + p1111}];
hZU = EntropyExpr[{p0000 + p0100 + p1000 + p1100, p0001 + p0101 + p1001 + p1101, p0010 + p0110 + p1010 + p1110, p0011 + p0111 + p1011 + p1111}];
hXYZ = EntropyExpr[{p0000 + p0001, p0010 + p0011, p0100 + p0101, p0110 + p0111, p1000 + p1001, p1010 + p1011, p1100 + p1101, p1110 + p1111}];
hXYU = EntropyExpr[{p0000 + p0010, p0001 + p0011, p0100 + p0110, p0101 + p0111, p1000 + p1010, p1001 + p1011, p1100 + p1110, p1101 + p1111}];
hXZU = EntropyExpr[{p0000 + p0100, p0001 + p0101, p0010 + p0110, p0011 + p0111, p1000 + p1100, p1001 + p1101, p1010 + p1110, p1011 + p1111}];
hYZU = EntropyExpr[{p0000 + p1000, p0001 + p1001, p0010 + p1010, p0011 + p1011, p0100 + p1100, p0101 + p1101, p0110 + p1110, p0111 + p1111}];
hXYZU = EntropyExpr[{p0000, p0100, p0001, p0101, p0010, p0110, p0011, p0111, p1000, p1100, p1001, p1101, p1010, p1110, p1011, p1111}];
Score2 = hZU - (hXZ - hX) - (hYU - hY); (* the score valid for (Subscript[ℒ, 2]) *)

f = Simplify[Score2 //. P]

Plot[f, { γ, 0, 1 }]
sol = FindInstance[f == 0 && 0 < γ < 1, {γ}, Reals, 10]; (* Ask for 10 solutions ... *)
{ Length[sol], sol[[1]], N[sol[[1]]]} (* ... but there exists only one root and it is far away from 0. *)

(* The non-Ingleton score is positive for γ > 0.7276. We can prove algebraically that these parameters do not yield a probability distribution under the given parametrization. *)
sol = FindInstance[γ > 0.727 && And @@ Map[#[[2]] ≥ 0 &, P], {γ, ϵ, δ123, δ124}, Reals]