A binomial-hypergeometric identity for graph-count coefficients

For integers $g\geq 1$, $\ell\geq 0$, and $j\geq 1$, let $\binom{n}{r}$ denote the binomial coefficient, let ${}_2F_1$ denote the Gauss hypergeometric function, and let the product over $m$ be taken for $0\leq m\leq j-1$. The binomial-hypergeometric identity. The following identity is conjectured:

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

The authors verified the identity for $0\leq\ell\leq20$ and $1\leq g,j\leq20$ and leave a general proof for future analysis; it arises by comparing two formulations of the graph counts ${\mathcal N}_{4,z}(g,j)$, although equality of the resulting linear combinations does not itself require equality of their coefficients.