Inverse images of positive closed currents under holomorphic endomorphisms of compact K\"ahler manifolds

We prove that for a surjective holomorphic endomorphism $f$ of a compact Kähler manifold $X$ of dimension $k\ge 2$ and for some integer $p$ with $1\le p\le k$, there exists a proper invariant analytic subset $E$ for $f$ such that if a positive closed $(p, p)$-current $S$ can be represented by a smooth form in a neighborhood of $E$, the sequence $d_p^{-n}(f^n)^*(S-\alpha_S)$ converges to $0$ exponentially fast in the sense of currents, where $d_p$ is the dynamical degree of order $p$ and $\alpha_S$ is a smooth closed $(p, p)$-form in the de Rham cohomology class of $S$.