Dubrovin's refined conjecture for Fano varieties

Let $X$ be a Fano variety of complex dimension $D$ with odd-vanishing cohomology, and let $F_0(\textbf{v})$ be a power series with convergence domain $\Omega$. Let $\mathcal{D}^b(X)$ denote the bounded derived category of coherent sheaves on $X$, and let $\chi(E_j,E_k)$ be the Euler pairing of objects in this category. Write $Y_R$ for the relevant solution and $Y_{\rm top}=\Psi\cdot\mathcal{Y}_{\rm top}$ for the topological-enumerative solution. Let $\widehat\Gamma^-_X$ be the negative Gamma class, let $c_1(X)$ be the first Chern class, and put $\bar D=D\ \text{mod}\ 2$. Dubrovin's refined conjecture. The quantum cohomology of $X$ is semisimple if and only if $\mathcal{D}^b(X)$ admits a full exceptional collection. If the quantum cohomology of $X$ is semisimple, then there exists a full exceptional collection $(E_1,\dots,E_n)$ such that the Stokes matrix $S$ is the inverse of the Euler matrix $\left(\chi(E_j,E_k)\right)_{1\leq j,k\leq n}$, and the central connection matrix $C$ connecting $Y_R$ with $Y_{\rm top}$ is the matrix whose columns are the coordinates of

$$\frac{i^{\bar D}}{(2\pi)^\frac{D}{2}}\widehat\Gamma^-_X\cup\exp(-\pi i c_1(X))\cup{\rm Ch}(E_k),\qquad 1\leq k\leq n.$$

This refines Dubrovin's original conjecture by specifying the central connection matrix through Gamma classes and Chern characters. The paper proves the conjecture for the Lagrangian Grassmannian $LG(2,4)$, while the general statement remains open.