Temporal regularity of marginal matrix-product growth

For a nonempty compact $\mathcal M \subset \mathbb{R}^{d \times d}$ with joint spectral radius $\widehat\rho(\mathcal M) = 1$, let $$g_{\mathcal M}(k) = \max_{A_1, \dots, A_k \in \mathcal M} \|A_k \cdots A_1\|_2$$ be the largest spectral norm of a product of length $k$. Following Varney and Morris, call $g_{\mathcal M}$ weakly increasing if there is $c > 0$ with $g_{\mathcal M}(k + \ell) \ge c\, g_{\mathcal M}(k)$ for all $k, \ell \ge 1$, and weakly upper regularly varying if for every $\ell \ge 1$ there is $C_\ell > 0$ with $g_{\mathcal M}(\ell k) \le C_\ell\, g_{\mathcal M}(k)$ for all $k \ge 1$. Varney and Morris proved that $g_{\mathcal M}(k + 1) \ge c\, g_{\mathcal M}(k)$ for some $c > 0$ and all $k \ge 1$, and that $g_{\mathcal M}$ is weakly upper regularly varying if, for some real invertible $X$ and some $1 \le j < d$, every $X A X^{-1}$ with $A \in \mathcal M$ is block upper triangular with diagonal blocks of sizes $j$ and $d - j$, and the upper-left blocks and the lower-right blocks each generate bounded sets of products. They also showed that if $\mathcal M$ is finite and $g_{\mathcal M}$ has both properties, then $g_{\mathcal M} / g_{\mathcal N}$ is bounded above and below by positive constants for some set $\mathcal N$ of two real matrices with $\widehat\rho(\mathcal N) = 1$.

Problem. Is $g_{\mathcal M}$ both weakly increasing and weakly upper regularly varying, for every $d \ge 1$ and every nonempty compact $\mathcal M \subset \mathbb{R}^{d \times d}$ with $\widehat\rho(\mathcal M) = 1$?