One-descent conjecture for generalized monotone triangles

Let $A_n:=\alpha(n;1,2,\ldots,n)$, and let $n\geq2$ and $i=1,\ldots,n-1$. One-descent conjecture. The identity

$$A_n=\alpha(n+2;1,2,\ldots,i+1,i,i+1,\ldots,n)$$

should hold. This is a computational conjecture related to a broader reverse-duplication identity developed later in the paper; the source does not provide a general proof.