Lin and Yan's zero-refinement conjecture for 120-avoiding inversion sequences

Let $cmathbf{I}_n$ be the set of inversion sequences of length $n$, and let $cmathcal{PA}_n$ be the set of primitive ascent sequences of length $n$. For an inversion sequence $e$, let $cmathrm{zero}(e)$ denote its number of zero entries. Let $c_{n,k}$ be defined by

$$c_{1,1}=1,\qquad c_{n,0}=0,$$

for $n\geq1$, and

$$c_{n,k}=c_{n-1,k-1}+k\sum_{j=k}^{n-1}c_{n-1,j}$$

for $n\geq2$ and $1\leq k\leq n$. Lin and Yan's zero-refinement conjecture. For $n\geq1$ and $1\leq k\leq n$,

$$\left|\left\{e\in\mathbf{I}_n(\underline{12}0):\operatorname{zero}(e)=k\right\}\right|=c_{n,k}.$$

The unrefined class of $coverline{12}0$-avoiding inversion sequences is known to be counted by the powered Catalan numbers; the conjecture asks for the refinement by the number of zero entries.