Sparsity conjecture for post-critically finite maps with fixed tail-length

Work over an algebraically closed field $\mathbb{F}$ of characteristic $0$. Let $\operatorname{End}_d^n$ denote the space of degree-$d$ endomorphisms of $\mathbb{P}^n$. For a map $f\in\operatorname{End}_d^n$, let $\operatorname{\mathcal{C}}_f$ be its critical locus. A map is post-critically finite of Type $(k,\ell)$ if $k\ge1$ and $\ell\ge0$ are minimal such that

$$f^{k+\ell}(\operatorname{\mathcal{C}}_f)\subseteq f^\ell(\operatorname{\mathcal{C}}_f).$$

Sparsity conjecture. Let $d\ge3$ and $n\ge2$. For every fixed tail-length $\ell\ge1$, the set

$$\left\{f\in\operatorname{End}_d^n:\text{$f$ is post-critically finite of Type $(k,\ell)$ for some $k\in\mathbb{N}$}\right\}$$

is contained in a proper Zariski closed subset of $\operatorname{End}_d^n$. The paper also asks whether the analogous assertion holds for the union over all tail-lengths.