Ehrhart positivity conjecture for generalized Gelfand–Tsetlin polytopes

Let $\mathcal{GT}(\lambda,\sigma)$ be the generalized Gelfand–Tsetlin polytope associated with a partition $\lambda$ and a permutation $\sigma$, and let $i(\mathcal{GT}(\lambda,\sigma),k)$ denote its Ehrhart polynomial. In the new variables $\lambda_i=a_1+a_2+\dotsb+a_i$, consider the polynomial in $k$ and the $a_i$. Ehrhart positivity conjecture. The Ehrhart polynomial $i(\mathcal{GT}(\lambda,\sigma),k)\in\mathbb{Q}[k]$ has only non-negative coefficients for all $\lambda$ and $\sigma$. Furthermore, $i(\mathcal{GT}(\lambda,\sigma),k)\in\mathbb{Q}[k,a_1,a_2,\dotsc,a_n]$ has non-negative coefficients. The conjecture predicts coefficientwise positivity both in the Ehrhart variable and after reparametrizing the highest-weight data by the successive parameters $a_i$; the paper presents this as an expectation rather than a proved theorem.