Total integrability of the pentagram map on spiral configuration spaces

Let ${\cal C}(n,k)$ be the space of configurations of pentagram spirals of type $(n,k)$, and let $T_{n,k}$ denote the pentagram map acting on this space.

Integrability conjecture. The map $T_{n,k}$ acting on ${\cal C}(n,k)$ is a discrete totally integrable system.

This is the proposed integrability principle for pentagram spirals, extending the integrable behavior known for the ordinary pentagram map. The supplied text presents it as a computer-experiment-driven conjecture, with no resolution stated.