A formula for a repeated-entry generalized monotone-triangle enumeration

Let $A_{n-1}:=\alpha(n-1;1,2,\ldots,n-1)$ and let $n\geq4$. Repeated-entry formula conjecture. The identity

$$\alpha(n+1;1,3,4,3,4,5,\ldots,n)=\frac{n+4}{2}A_{n-1}$$

should hold. The formula was obtained from computational experiments and is presented as part of a family of conjectured evaluations; no proof is given in the source.