Double Hurwitz polynomial-structure conjecture

Let $F_{g,n}$ be the free-energy generating functions associated with the double Hurwitz numbers, let $a_1,\ldots,a_d$ be the branch points of the spectral curve, and let $\sigma_i$ be the local involution associated with $a_i$. Double Hurwitz polynomial-structure conjecture. For $2g-2+n>0$, $F_{g,n}(z_1,\ldots,z_n)$ is rational with poles only at $z_i=a_j$ for $i=1,\ldots,n$ and $j=1,\ldots,d$, and

$$F_{g,n}(z_1,z_2,\ldots,z_n)+F_{g,n}(\sigma_i(z_1),z_2,\ldots,z_n)$$

is analytic at $z_1=a_i$ for $i=1,\ldots,d$. These constraints are called the linear loop equations. They are proposed as the polynomial-like structure underlying the main topological-recursion conjecture; the source states that the conjectures are equivalent, but gives no general proof or resolution.