Entringer number conjecture for 000-avoiding inversion sequences

The Entringer numbers $d_{n,k}$ count down/up permutations of $[n+1]$ whose first entry is $k+1$, and satisfy

$$d_{n,k}=d_{n,k-1}+d_{n-1,n-k}.$$

An Entringer number conjecture asserts that, for $n\geq 1$ and $0\leq k\leq n-1$, $d_{n,k}$ is the number of inversion sequences $e\in\mathbf{I}_n(000)$ with last entry $e_n=k-1$.

This conjecture proposes a direct interpretation of the Entringer numbers in terms of 000-avoiding inversion sequences; the source reports it as suggested by computations, and no resolution is provided.