Bijection between constrained Delannoy paths and Catalan paths avoiding symmetric peaks

For a positive integer $n$, let $E\mathcal{P}^{(1)}_{n,n}(NE,EN)$ be the set of Delannoy paths from $(0,0)$ to $(n,n)$ consisting of north-steps $N=(0,1)$, east-steps $E=(1,0)$, and diagonal-steps $D=(1,1)$ that avoid the patterns $NE$ and $EN$, end with an east step $E$, and are not below $y=x$. Let the second set consist of Catalan paths of size $n+1$ avoiding symmetric peaks, where a symmetric peak is a peak whose maximal mountain has the form $N^iE^i$.

Delannoy–Catalan bijection conjecture. For every positive integer $n$, there exists a bijection between $E\mathcal{P}^{(1)}_{n,n}(NE,EN)$ and the set of Catalan paths of size $n+1$ avoiding symmetric peaks.

This conjecture proposes a direct combinatorial correspondence between a class of constrained Delannoy paths and Catalan paths with a local peak-avoidance condition. The surrounding discussion identifies the related generating-function identity, but the supplied text gives no evidence that this bijection has been proved or disproved.