Galkin's lower bound conjecture for the spectral radius of quantum multiplication

For a Fano manifold $X$, let $QH^\bullet(X):=H^{\rm ev}(X)\otimes_{\mathbb{Q}}\mathbb{Q}[\mathbf{q}]$ be its even quantum cohomology, and let $\hat c_1$ be quantum multiplication by $c_1(X)$ evaluated at $\mathbf{q}=1$. Write $\rho(\hat c_1)$ for the spectral radius of this operator. Galkin's lower bound conjecture.

$$\rho(\hat c_1)\geq \dim X+1,$$

with equality if and only if $X$ is isomorphic to a projective space $\mathbb{P}^n$.

Here and throughout, the dimension is the complex dimension. The conjecture has been verified in several cases, including complex Grassmannians, Lagrangian and orthogonal Grassmannians, and Fano complete intersections in projective spaces; its general status is not resolved in the supplied source.