Even-argument reciprocity conjecture for generalized monotone triangles

Let $\alpha(n;k_1,\ldots,k_n)$ denote the generalized monotone-triangle enumeration, and let $n\geq1$. Even-argument reciprocity conjecture. The identity

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

should hold. The paper lists this among identities supported by mathematical computation and explains that the combinatorial interpretation supplied by generalized monotone triangles makes bijective proofs desirable; no proof is supplied here.