The r-ELSV conjecture for r-spin Hurwitz numbers

Let $r$ be a positive integer. For integers $k_1,\dots,k_n$, write

$$k_i=rp_i+(r-1-a_i),$$

where $p_i$ is the quotient and $r-1-a_i$ is the remainder on division by $r$, and set

$$m=\frac{\sum_i k_i+n+2g-2}{r}.$$

Let $h_{g,r;k_1,\dots,k_n}$ be the stated relative stable-map Gromov–Witten invariant, and let $f_{g,r;k_1,\dots,k_n}$ be the stated integral over the moduli space of $r$-spin structures. The $r$-ELSV conjecture.

$$h_{g,r;k_1,\dots,k_n}=f_{g,r;k_1,\dots,k_n}.$$

The conjecture proposes an equality between completed-cycle Hurwitz-type Gromov–Witten invariants and $r$-spin intersection integrals. The source says it is proved in genus $0$ and for $(g,n)=(1,1)$ with arbitrary $r$ (the latter unpublished), while the general statement remains open.