Stability conjecture for holomorphic maps with multiple degree

Let $X$ be a nilpotent complex algebraic variety such that $\pi_2(X)$ is a free abelian group of rank $r$. Assign to each holomorphic map $f:\mathbb{C}\mathbb{P}^1\to X$ a multiple degree $\mathbf{n}$, and let $g_{\mathbf{n},d\mathbf{n}}$ be the injective map obtained by precomposition with the degree-$d$ self-cover of $\mathbb{C}\mathbb{P}^1$. Stability conjecture for multiple degree. Let $X$ be a variety as above. Then the map $g_{\mathbf{n},d\mathbf{n}}$ induces a rational homotopy equivalence up to dimension $k_{\mathbf{n}}$, where $k_{\mathbf{n}}$ grows with $\mathbf{n}$ as $\mathbf{n}$ moves to infinity in a suitable positive cone. This extends the proposed stability property from a single degree to multiple degrees; the source does not provide evidence resolving it.