Equidistribution of generic analytic subsets toward powers of the Green current

Let $f$ be a holomorphic endomorphism of $\mathbb{P}^k$ of algebraic degree $d\geq 2$, and let $T$ be its Green current. For an analytic subset $H$ of $\mathbb{P}^k$ of pure codimension $p$ and degree $s$, call $H$ generic if either $H\cap E=\varnothing$ or

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

for any irreducible component $E$ of every totally invariant analytic subset of $\mathbb{P}^k$.

Equidistribution conjecture. For every generic $H$, one has

$$d^{-pn}(f^n)^*[H]\longrightarrow sT^p.$$

The claim is an analogue of equidistribution toward the Green current for higher-codimension analytic sets. The source states that the authors proved this conjecture in the cited work, so it is recorded as solved.