Refined Dubrovin conjecture and Gamma-conjecture II for Fano varieties

Let $X$ be an $n$-dimensional smooth Fano variety satisfying $h^{p,q}(X)=0$ for every $p\ne q$. Let $\widetilde\mu=\dim_{\mathbb C}\bigoplus_{p\in\mathbb Z}H^p(X;\mathbb C)$, let $\widehat\Gamma_X=\prod_i\Gamma(1+\delta_i)$ be the Gamma class in terms of the Chern roots $\delta_i$ of $TX$, and let $\operatorname{Ch}(\mathcal E)$ denote the modified Chern character. The refined Dubrovin conjecture and Gamma-conjecture II. (1) The quantum cohomology of $X$ is semisimple if and only if $D^b\operatorname{Coh}(X)$ has a full exceptional collection. (2) If the quantum cohomology is semisimple, then for every oriented line $\ell$ with $\phi\in[0,1)$ there is a correspondence between monodromy data $(S^\phi,C^\phi)$ and full exceptional collections $(\mathcal E_1^\phi,\dots,\mathcal E_{\widetilde\mu}^\phi)$. (3) For the corresponding collection, $S^\phi_{ij}=\chi(\mathcal E_i^\phi,\mathcal E_j^\phi)$, and the $j$-th column of $C^\phi$ is

$$C_j^\phi=\frac{1}{(2\pi)^{n/2}}\widehat\Gamma_X\operatorname{Ch}(\mathcal E_j^\phi).$$

These conjectures relate semisimplicity and exceptional collections to the monodromy data of quantum cohomology. The source presents them as conjectural refinements; no resolution is supplied.