Ascent conjecture for Schröder paths and 021-avoiding inversion sequences

Let $R_{n-1}$ denote the Schröder paths of length $n-1$, and let an ascent be a maximal sequence of consecutive up steps in a Schröder path. Let $\mathbf{I}_n(021)$ denote the inversion sequences of length $n$ avoiding the pattern $021$.

Ascent conjecture. The number of Schröder paths $p\in R_{n-1}$ with $k-1$ ascents is equal to the number of inversion sequences $e\in\mathbf{I}_n(021)$ with $k$ distinct values.

This conjecture predicts an equidistribution between ascents in Schröder paths and the number of distinct values in 021-avoiding inversion sequences. The source says that the correspondence is suggested by computations and that the previously constructed bijections do not establish it; no resolution is provided.