Zvonkine's topological-recursion conjecture for orbifold $r$-spin Hurwitz numbers

Let $g\geq 0$ and $n\geq 1$. For positive integers $\mu_1,\ldots,\mu_n$, let $h_{g;\mu}^{\circ,q,r}$ be the connected $q$-orbifold $r$-spin Hurwitz number, and define the formal symmetric $n$-differential

$$W_{g,n}(x_1,\ldots,x_n)=d_1\otimes\cdots\otimes d_n\sum_{\mu_1,\ldots,\mu_n=1}^{\infty}h_{g;\mu}^{\circ,q,r}\prod_{i=1}^n x_i^{\mu_i}.$$

Let $\omega_{g,n}(z_1,\ldots,z_n)$ be symmetric $n$-differentials on the spectral curve with parametrization $x(z)=ze^{-z^{qr}}$ and $y(z)=z^q$. Zvonkine's topological-recursion conjecture. The formal differentials $W_{g,n}$ are expansions in $x_1,\ldots,x_n$ of the differentials $\omega_{g,n}$, and the latter satisfy topological recursion on this spectral curve.