Dinh–Sibony equidistribution conjecture for generic analytic subsets of projective space

Let $0\le s\le k$ be an integer. Let $f:\mathbb{P}^k\to\mathbb{P}^k$ be a holomorphic endomorphism of degree $d\geq 2$, and let $T$ be its Green $(1,1)$-current. For an analytic subset $H$ of $\mathbb{P}^k$ of pure dimension $k-s$ and degree $c_H$, let $[H]$ denote the current of integration on $H$. The subset $H$ is generic if either $H\cap\mathcal{E}=\emptyset$ or

$$\operatorname{codim}(H\cap\mathcal{E})=s+\operatorname{codim}(\mathcal{E})$$

for any irreducible component $\mathcal{E}$ of every totally invariant analytic subset of $\mathbb{P}^k$. Dinh–Sibony's conjecture. For every generic $H$, the sequence

$$d^{-sn}(f^n)^*[H]$$

converges to $c_HT^s$. This conjecture extends known equidistribution results for non-invertible endomorphisms of projective space from currents satisfying additional regularity or integrability conditions to generic analytic subsets. Its status is not determined in the supplied context.