Borot–Garcia-Failde's exchange conjecture for fully simple map amplitudes

Let $\mathcal{C}$ be the spectral curve for ordinary maps or maps with loops, with coordinate $x$ and disk amplitude $w=W_{1}^{[0]}(x)$. Let $\check{\omega}_{n}^{[g]}$ be the topological-recursion amplitudes for the initial data obtained by exchanging $x$ and $w$, and let $X_{n}^{[g]}$ denote the generating series of fully simple maps. For $2g-2+n>0$, the variables $z_i$ are points of $\mathcal{C}$ and the equality is understood as a formal Laurent-series identity near $z_i\to[\infty]$.

Fully simple amplitude conjecture. For ordinary maps or maps with loops,

$$X_{n}^{[g]}(w(z_1),\ldots,w(z_n))=\frac{\check{\omega}_{n}^{[g]}(z_1,\ldots,z_n)}{\mathrm{d}w(z_1)\cdots\mathrm{d}w(z_n)},\qquad 2g-2+n>0.$$

This is the enumerative form of the proposed $x\leftrightarrow w$ symplectic exchange. The disk and cylinder cases are proved in the source, while the general stable-topology statement remains conjectural.