The hypermap topological-recursion conjecture

For a positive integer $a$, let $M_{g,n}^{[a]}(b_1,\ldots,b_n)$ denote the enumeration coefficients of $a$-hypermaps, and let $\omega_{g,n}$ be the correlation differentials produced by topological recursion on the rational spectral curve

$$x(z)=z^{a-1}+\frac{1}{z},\qquad y(z)=z.$$

Write $x_i=x(z_i)$. Hypermaps topological-recursion conjecture. For a fixed positive integer $a$, the correlation differentials have expansions at $x_i=\infty$ satisfying

$$\omega_{g,n}=\sum_{b_1, b_2, \ldots, b_n=1}^{\infty}M_{g,n}^{[a]}(b_1,b_2,\ldots,b_n)\prod_{i=1}^n\frac{b_i}{x_i^{b_i+1}}\\,\mathrm{d}x_i,\qquad 2g-2+n>0.$$

This conjecture would relate the enumeration of $a$-hypermaps to topological recursion through the semiclassical limit of their quantum curve. The source reports considerable numerical evidence, but leaves the general statement conjectural.