Borel summability conjecture for transseries solutions of Painlevé III'

Let $R_{\rm odd}$ be the odd part of the formal WKB solution, and let $\lambda(t,{\bf c},\eta;\alpha)$ be the corresponding transseries solution of $(P_{\rm III'})_{D_{6}}$, with formal power-series components $\lambda^{(k)}(t,{\bf c},\eta)$. Assume that the integral of $R_{\rm odd}$ in the first part of the one-parameter solution is taken along a path that never touches any turning points, the simple pole, or Stokes curves, and assume that $\operatorname{Re}\phi<0$. Borel summability conjecture. The transseries solution is Borel summable in the general sense, each $\lambda^{(k)}(t,{\bf c},\eta)$ is Borel summable, and

$${\cal S}[\lambda(t,{\bf c},\eta;\alpha)] = \sum_{k \ge 0} (\alpha \eta^{-1/2})^{k} {\cal S}[\lambda^{(k)}(t,{\bf c},\eta)] e^{k\eta\phi}$$

converges for sufficiently large $\eta>0$ and represents an analytic solution of $(P_{\rm III'})_{D_{6}}$. Here ${\cal S}[\lambda^{(k)}(t,{\bf c},\eta)]$ denotes the Borel sum of the formal power series $\lambda^{(k)}(t,{\bf c},\eta)$. The conjecture concerns the Borel summability of transseries solutions under the stated Stokes-geometric conditions; the paper motivates it by a result of Kamimoto, but the supplied text does not establish the full assertion.