The Laplace-transform conjecture for the J-function

Let $\mathcal C=(\Sigma,x,y)$ be a spectral curve and let $\gamma\subset\Sigma\setminus{\bf b}$ be an admissible path associated with $e^{-x/\mathfrak u}$. Let $\eta$ be the CohFT pairing, let $\Phi(\gamma,-\mathfrak u)$ be the class associated with $\gamma$, and let $S^*$ be the dual calibration operator. Laplace-transform conjecture. There exists a $\nu$-vector $\nu(\mathfrak u)$ such that the J-function $J(-\mathfrak u)=-\mathfrak u S^*(-\mathfrak u)\nu(\mathfrak u)$ satisfies

$$\int_\gamma e^{-x(z)/\mathfrak u}y(z)\,dx(z)=-\eta\bigl(J(-\mathfrak u),\Phi(\gamma,-\mathfrak u)\bigr).$$

This conjecture identifies the spectral-curve one-point function with the geometric J-function through a Laplace transform. The paper checks it in the two examples studied, but does not establish it in general.