Hypergeometric rationality conjecture for unicellular Ising maps with many monochromatic edges

Let $U_{n,k,\ell}$ denote the coefficient of $t^{3n}\nu_{\bullet}^k\nu_{\circ}^{\ell}$ in the generating series $U$ for unicellular Ising maps. For fixed $i,j\geq 0$, the coefficient $U_{n,3n-i,j}$ is considered with $n=2g-1$.

Many-monochromatic-edges conjecture. For fixed $i,j\geq 0$, there exists a rational function $R_{i,j}(g)$ in $g$ such that

$$U_{n,3n-i,j}=R_{i,j}(g)\frac{(6g)!}{12^g g!(3g)!}.$$

These numbers vanish when $i+j$ is not a multiple of $3$ or when $j>i$.

The conjecture extends the known hypergeometric formula for monochromatic unicellular cubic maps. The case $i-j=1$ is proved using a decomposition along the unique bicolored edge and known results on rooted precubic maps, while the general case remains open.