Khesin-Soloviev integrability conjecture for diagonal generalized pentagram maps

Let $(v_i)$ be a twisted polygon in $b2b2\text{?}b2b2$ and let $I$ be a multi-index defining a generalized pentagram map $T_{I,J}$. The map $T_{I,I}$ is the diagonal case in which the two multi-indices coincide. Khesin-Soloviev conjecture. The maps $T_{I,I}$ are integrable for all multi-indices $I$. The source reports numerical evidence for some cases and established integrability for others, but does not state that the general conjecture is resolved.