Reverse-duplication conjecture for generalized monotone triangles

Let $A_n:=\alpha(n;1,2,\ldots,n)$, and let $n\geq1$, $i=1,\ldots,n-k+1$, and $k=1,\ldots,n$. Reverse-duplication conjecture. The identity

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

should hold. In words, the consecutive subsequence $(i,i+1,\ldots,i+k-1)$ is reversed, each entry is duplicated, and the resulting block is inserted back into $(1,2,\ldots,n)$. The paper notes that the previously stated identity is the case $k=2$, while the earlier combinatorial-reciprocity identity is the case $k=n$; the general statement remains computationally conjectural.