Galkin-Golyshev-Iritani Conjecture O for Fano manifolds

Let $X$ be a Fano manifold. Consider the even cohomology $H^\bullet(X):=H^{\rm even}(X)$ and the operator $\hat c_1(X)$ given by quantum multiplication by $c_1(X)$, evaluated at all quantum parameters equal to $1$. Let $\rho=\rho(\hat c_1(X))$ be its spectral radius, and let $s$ be the Fano index of $X$:

$$s=\max\left\{k\in\mathbb{Z}:\frac{c_1(X)}{k}\in H^2(X,\mathbb{Z})\right\}.$$

Galkin-Golyshev-Iritani's Conjecture O. Every Fano manifold $X$ satisfies: $\rho$ is an eigenvalue of $\hat c_1$ with multiplicity one; and for every eigenvalue $\lambda$ of $\hat c_1$ satisfying $|\lambda|=\rho$,

$$\lambda^s=\rho^s.$$

Conjecture O has been verified in various cases, including the two blow-ups studied in the paper, but the supplied source does not assert a general resolution.