6 problems

Rank-$n+1$ hypergraph conjecture. If an $n$-party inequality contracts on $(n+1)$-graphs, then it is a valid inequality on all hypergraphs of finite rank.

The entropy polynomial root conjecture. The polynomial $p_{k,r}(x)$ has exactly two real roots in $(0,1)$, counting multiplicity. The paper explains that this root statement would…

The entropy inequality for real exponents. For $0\leq x\leq 1$,

Fix $\alpha>0$ and $s\geq1$. Let $U$ be the affine-group representation on $L^2(\mathbb{R}_+)$, let $\eta_\alpha(k)=C(\alpha)k^\alpha e^{-k}$, and define…

Let $X_1$ and $X_2$ be binary classical random variables with quantum side-information systems $B_1$ and $B_2$, respectively, and let $\rho^{X_1B_1}$ and $\rho^{X_2B_2}$ be classic…

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