Sibuya-type conjecture for the third-order conformal-limit equation

Let $M>0$, and consider the third-order equation

$$\left[\partial_x^3-g(g+2)\left(\frac{1}{x^2}\partial_x-\frac{1}{x^3}\right)+p(x,E)\right]y(x,E,g)=0.$$

Here $p(x,E)$ is the potential in the conformal-limit equation, and $x$ is understood on a suitable cover of the punctured complex plane when the branch point at $x=0$ requires it. Sibuya-type conjecture. Equation above has a solution $y(x,E,g)$ that is entire in $(x,E)$, admits for $M>1/2$ the asymptotic representations

$$y\sim x^{-M}e^{-\frac{1}{M+1}x^{M+1}},\qquad y'\sim-e^{-\frac{1}{M+1}x^{M+1}},\qquad y”\sim x^{-M}e^{-\frac{1}{M+1}x^{M+1}},$$

as $x\to\infty$ in the sector

$$|\arg(x)|<\frac{4\pi}{3M+3},$$

and is uniquely fixed by these properties. This proposed generalization of Sibuya's theorem is motivated by the ODE/conformal-field-theory correspondence; the supplied text does not establish whether the claim is proved or remains open.