Bijection between constrained Delannoy paths and 102-avoiding inversion sequences

Let $n$ be a positive integer. Let $D\mathcal{P}^{(2)}_{n,2n}(NE,EN)$ be the set of Delannoy paths from $(0,0)$ to $(n,2n)$ 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 a diagonal step $D$, and are not below $y=2x$. Let $\mathcal{IS}_n(102)$ denote the set of inversion sequences of length $n$ avoiding the pattern $102$. Bijection conjecture. For every positive integer $n$, there exists a bijection between $D\mathcal{P}^{(2)}_{n,2n}(NE,EN)$ and $\mathcal{IS}_n(102)$. The result would provide a combinatorial interpretation of the generating function $F_D^{(2)}(x)$, which the source identifies with the generating function of $\mathcal{IS}_n(102)$; the source gives no resolution of the asserted bijection.