Self-replication conjecture for the formal Baker–Akhiezer kernel

Let ${\cal S}$ be a spectral curve, let $\psi({\cal S},\hbar;p_1,p_2)$ be its formal Baker–Akhiezer kernel, and define $\Omega_p(q)=\frac{B(p,q)}{dx(p)}$ and $\delta_p=dx(p)\,\partial_{\Omega_p}$. Self-replication conjecture. Under assumptions omitted in the source, the formal Baker–Akhiezer kernel should satisfy

$$\delta_p\psi({\cal S},\hbar;p_1,p_2)=-\psi({\cal S},\hbar;p_1,p)\,\psi({\cal S},\hbar;p,p_2).$$

The source states that this is equivalent to the Hirota equation for the Tau function and implies determinantal formulae, Plücker relations, and an isomonodromic integrable system with an operator $L_q(\hbar,x)\in{\cal M}({\cal C}_0)[[\hbar]]$ satisfying $(d_p-L_q(\hbar,x(p)))\psi({\cal S},\hbar;p,q)=0$; the assumptions and precise scope are deferred to the cited work.