Periodic spectral-radius attainment for linear cocycles with closing by periodic orbits

Let $T\colon W\rightarrow W$ have the closing by periodic orbits property, and let $\boldsymbol{C}\in\operatorname{C}(W,\mathbb{R}^{d\times d})$ be a continuous matrix-valued function. Let $\mathscr{C}(n,w)=C(T^{n-1}(w))\dotsm C(w)$ be the induced linear cocycle and let $\boldsymbol{\rho}(\boldsymbol{C},T)$ denote its joint spectral radius. Periodic spectral-radius attainment conjecture. There should exist a periodic point $w^*$ of $T$ of period $n^*$, for some $n^*\geq1$, such that

$$\boldsymbol{\rho}(\boldsymbol{C},T)=\sqrt[n^*]{\rho(\mathscr{C}(n^*,w^*))}.$$

This asks whether the joint spectral radius is always realized by a single periodic orbit under the closing-by-periodic-orbits hypothesis. The question concerns spectral finiteness for linear cocycles and is relevant to optimization control, wavelets, numerical computation of spectral radii, and random matrices; the supplied text gives no resolution.