Painlevé I limit conjecture for the rescaled tau-function near the critical surface

The critical surface is parameterized by

$$\left\{(t_1(\varsigma,\eta),t_2^{\pm}(\varsigma,\eta),t_5(\varsigma,\eta)\mid \eta\in \operatorname{\mathbb{R}},\ \varsigma > \max\left\{\frac{5}{3}\eta,0\right\}\right\},$$

with

$$t_1(\varsigma,\eta) = -\frac{5}{12}\varsigma\left(5\eta-9\eta\varsigma+3\varsigma^2\right),\qquad t_2^{\pm}(\varsigma,\eta) = \pm\frac{\sqrt{2\varsigma}}{12}(5\eta-3\varsigma)^2,\qquad t_5(\varsigma,\eta) = \eta.$$

Let $P_0 = \langle t_1^{(0)},t_2^{\pm (0)},t_5^{(0)}\rangle$ be a point on this surface, and let ${\bf u} = \langle u_1,u_2,u_5\rangle$ be a unit vector based at $P_0$ lying in the region $D$ below the tangent plane of the critical surface. Define

$$X_1(x|T):= T^6 t_1^{(0)} + C T^{2/5}u_1 x,\qquad X_2(x|T):= T^5 t_2^{\pm (0)} + C T^{-3/5}u_2 x,\qquad X_5(x|T):= T^2 t_5^{(0)} + C T^{-18/5}u_5 x.$$

Painlevé I limit conjecture. There exists a choice of constant $C = C(P_0,{\bf u})>0$ and a polynomial $Q(t_1,t_2,t_5)$ such that, defining $\hat{\tau}_0:= e^{Q}$, considered as a differential in $x$,

$$\lim_{T\to \infty} {\bf d} \log \frac{\tau(X_5(x|T),X_2(x|T),X_1(x|T))}{\hat{\tau}_0(X_5(x|T),X_2(x|T),X_1(x|T))} = -\mathcal{H}(x)dx,$$

where $\mathcal{H}(x)$ is the Painlevé I Hamiltonian. This asserts that a broad class of limits of the tau-function near the critical surface is governed by the Painlevé I Hamiltonian; the source provides no resolution status for this conjectural claim.