The hypergeometric identity conjecture for even-valence map counts

Let $g$, $\ell$, and $j$ be integers satisfying

$$g\geq 1,\qquad \ell\geq 0,\qquad j\geq 1.$$

Hypergeometric identity conjecture. The following identity should hold:

$$j!\,2^{\ell+2g-1}\binom{(2g-2)+(\ell+j)}{j}\,{}_2F_1\left(\genfrac{}{}{0pt}{}{-j,-2j}{2-2g-(\ell+j)};-1\right)=\sum_{k=1}^{\ell+2g}\binom{\ell+2g-1}{k-1}\prod_{m=0}^{j-1}2(2m+k).$$

The identity arose by comparing hypergeometric and explicit formulas for even-valence map counts. The source calls it a combinatorial conjecture and gives no resolution.