Determinantal formula for the Baker–Akhiezer kernel

Let $x$ be the spectral-curve coordinate and let $\widetilde{K}(z_1,z_2)$ be the formal Baker–Akhiezer kernel defined by

$$\widetilde{K}(z_1,z_2)=\frac{\exp\left(-N\int_{z_2}^{z_1}y\,dx\right)}{(z_1-z_2)\sqrt{x'(z_1)x'(z_2)}}\exp\left[\sum'_{g,l}\frac{N^{2-2g-l}}{l!}\underbrace{\int_{z_2}^{z_1}\cdots\int_{z_2}^{z_1}}_{l}\omega_l^{(g)}\right],$$

where the primed sum excludes all terms with $2-2g-l\geq 0$. Let $K(x_1,x_2)$ denote the determinantal kernel associated with the correlators $W_n^{(g)}$. Baker–Akhiezer kernel conjecture. The formal Baker–Akhiezer kernel equals the determinantal kernel after expressing its arguments through $x$:

$$\widetilde{K}(z_1,z_2)=K(x(z_1),x(z_2)).$$

If true, this identification would extend the determinantal formula already established for the one-point function to the full kernel and hence provide determinantal formulae for the correlators. The supplied text does not state whether the conjecture has been resolved.