Integral Stokes coefficient conjecture for complex geodesics and heat kernels

Let $(M,g)$ be real analytic, and let $(M_{\mathbb C},g_{\mathbb C})$ be a complexification satisfying the paper's assumption. Fix endpoints $x,y$, and let $\{\gamma_i\}_{i\in I}$ be the relevant no-conjugate holomorphic geodesics from $x$ to $y$. Define their phases by

$$\Phi_i(x,y):=\frac{\ell_{\gamma_i}(x,y)^2}{4}.$$

For each $\gamma_i$, consider a normalized formal heat solution

$$\widetilde K_i(\hbar;x,y)=(4\pi\hbar)^{-n/2}e^{-\Phi_i(x,y)/\hbar}\widetilde u_i(\hbar;x,y),$$

where $\widetilde u_i(\hbar;x,y)\in\mathbb C[[\hbar]]$ is $1$-Gevrey.

Integral Stokes coefficient conjecture. Each $\gamma_i$ determines such a normalized formal heat solution, and there exists a coefficient $S_{ij}^{+}\in\mathbb Z$ such that

$$\dot\Delta_{\Phi_j-\Phi_i}^{+}\widetilde K_i=S_{ij}^{+}\widetilde K_j.$$

This conjecture proposes that the Stokes coefficients governing alien derivatives are integers, reflecting signed counts of Picard–Lefschetz connecting trajectories. The statement is motivated by finite-dimensional exponential integrals, but its validity for the heat kernel remains open.