The KdV initial-condition formula for the non-perturbative descendent potential

Let $U({\bf p};\hbar)=\partial_{p_1}^2\log\bigl(\mathcal D^{\rm NP}_{\mu,\nu}(\hbar\cdot{\bf t}({\bf p});\hbar;w)\bigr)$ be the stated solution of the KdV equations. Let $G_2$, $\epsilon$, $\tau$, $w$, and $\theta\bigl[\begin{smallmatrix}\mu\\nu\end{smallmatrix}\bigr]$ have the meanings used in the paper. KdV initial-condition conjecture. The initial condition $U(p_1;\hbar)$ is

$$U(p_1;\hbar)=\frac{3G_2(\epsilon+\hbar p_1)^2}{4\pi\mathbf i}+\frac{\hbar^2}{8(\epsilon+\hbar p_1)^2}+\partial_{p_1}^2\log\theta\!\left[\begin{smallmatrix}\mu\\nu\end{smallmatrix}\right]\!\left(w-\frac{(\epsilon+\hbar p_1)^2}{2\sqrt{2\pi\mathbf i}\,\hbar};\tau\right).$$

In particular, for $\mu=\nu=1/2$,

$$U(p_1;\hbar)=\frac{G_2w\hbar}{\sqrt{2\pi\mathbf i}}+\frac{\hbar^2}{8(\epsilon+\hbar p_1)^2}+\partial_{p_1}^2\log\sigma\!\left(w-\frac{(\epsilon+\hbar p_1)^2}{2\sqrt{2\pi\mathbf i}\,\hbar};\tau\right).$$

This gives the datum determining the KdV solution in the paper's setting. The supplied context does not state whether this formula is proved or conjectural beyond its placement in a conjecture environment.