Singularity confinement conjecture for the pentagram map in even dimension

Let $n$ be even. Let $S$ be a type and let $X_S$ denote the corresponding stratum of configurations for the pentagram map $T$. Singularity confinement means that an iterate of $T$ becomes nonsingular at a generic point of $X_S$.

Singularity confinement conjecture. Singularity confinement holds generically on $X_S$ unless

$$[1,n-1]_2\subseteq S$$

or

$$[2,n]_2\subseteq S.$$

Whenever singularity confinement holds for a type, there exists an $m\leq n$ such that generic singularities of that type last $m$ steps; that is, $T^m$ is singular but $T^{m+1}$ is not.

For odd $n$, the preceding results establish an $(n+1)$-step bound outside the exceptional type, while this conjecture proposes the analogous statement for even $n$, with the indicated exceptional types. The claim about the exact confinement time strengthens the bound by requiring some $m\leq n$ for every type where confinement occurs.