Median-resummation conjecture for strong-coupling Stokes constants

Let $f_\infty(y)$ be the generating function in Eq., and let $\tilde{f}_\infty(y)$ be its asymptotic expansion as $y\to0$ with $\Im(y)>0$. Strong-coupling median-resummation conjecture. The median resummation reconstructs the original function:

$$\mathcal{S}^{\mathbf{med}}_{\frac{\pi}{2}}\tilde{f}_\infty(y)=f_\infty(y),\qquad y\in\mathbb{H}.$$

The claim concerns the effectiveness of median resummation for generating functions of strong-coupling Stokes constants and is supported in the source by numerical tests, while an analytic proof is stated to be missing.