The Laplace transformability conjecture for Borel-transformed WKB solutions

Let $\cQ$ be an $\hbar$-differential operator, lifted to an $\hbar$-connection after fixing a spin structure, and assume WKB-regularity. Let $L$ be the characteristic variety of $\cQ$, and let $x\in C$ lie outside the turning points. Consider the Borel transformation of a formal WKB solution at $x$. The Laplace transformability conjecture. The Borel transformation is Laplace transformable along any path going to $\infty$ avoiding poles.

This is the second conjectural condition in the paper's ideal higher-order spectral scattering picture. The surrounding discussion presents it as a further assumption after endless continuability, and the supplied text gives no evidence that it has been resolved.