Universal pentagram maps conjectures on Hamiltonianity and integrability

Let $T_{\mathcal I,\mathcal J}$ be a universal pentagram map on $d$-tuples of twisted $n$-gons in ${\mathbb {RP}}^d$, determined by matrices $\mathcal I$ and $\mathcal J$. Let $T_{I,J}$ denote a generalized pentagram map as in the paper's definition. Universal pentagram maps conjectures. (a) Every universal pentagram map $T_{\mathcal I,\mathcal J}$ is a discrete Hamiltonian system, meaning that it preserves a Poisson structure, although it need not be integrable. (b) A necessary condition for integrability of $T_{\mathcal I,\mathcal J}$ is that it be equivalent to a map $T_{I,J}$. These claims concern universal maps, which include classical and skew examples; Hamiltonianity is conjectured independently of integrability, while the stated equivalence is only a necessary condition for integrability.