The four-atom conjecture on the minimum Ingleton score

Given four random variables $A,B,C,D$ with a probability distribution, define the Ingleton score by

$$\frac{I(A;B|C)+I(A;B|D)+I(C;D)-I(A;B)}{H(ABCD)}.$$

Four-atom conjecture. The lowest possible Ingleton score is approximately $-0.089373$. It is attained by a four-variable distribution with alphabet size two, given by

$$P(0,0,0,0)=P(1,1,1,1)=\alpha$$ $$P(0,1,0,1)=P(0,1,1,0)=0.5-\alpha,$$

where $\alpha$ takes on a value approximately equal to $0.350457$.

This conjecture identifies a binary four-variable distribution that is claimed to minimize the Ingleton score, a normalized measure of violation of Ingleton's inequality. The parser provides no evidence resolving whether this minimum is indeed globally optimal.