Fischer–Riegler conjecture for vertically symmetric alternating-sign matrices

Let $\alpha(n;k_1,\ldots,k_n)$ denote the generalized monotone-triangle enumeration, and let $n\geq1$. An alternating-sign matrix (ASM) is a square matrix with entries in $\{-1,0,1\}$ whose row and column sums are $1$, with nonzero entries alternating in every row and column. Fischer–Riegler conjecture. The identity

$$\alpha(2n+1;2n+1,2n,\ldots,1)=(-1)^n\alpha(n;2,4,\ldots,2n)$$

should hold; moreover, $\alpha(n;2,4,\ldots,2n)$ counts vertically symmetric ASMs of size $2n+1$. This is one of several identities conjectured from computational evidence, and the source highlights the interest of finding bijective proofs.