Hypergeometric rationality conjecture for unicellular Ising maps with few monochromatic edges

Let $U_{n,i,j}$ denote the coefficient of $t^{3n}\nu_{\bullet}^i\nu_{\circ}^j$ in the generating series $U$ for unicellular Ising maps, and set $n=2g-1$.

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

$$U_{n,i,j}=Q_{i,j}(g)\frac{(2g)!(3g)!}{3^g g!^3}.$$

These numbers vanish when $i-j$ is not a multiple of $3$.

This conjecture concerns the opposite regime from maps with many monochromatic edges. The source gives several predicted instances, but no proof of the general rational-function form is provided.