Openness conjecture for extremal norms and characteristic words

Let $r,d \in \mathbb{N}$, let $\mathcal{I}_r(\mathbb{K}^d)$ be the ambient class of $r$-tuples under consideration, and let $\Omega_r^n$ denote the words of length $n$. For $\hat \omega \in \Omega_r^n$, define $\mathcal{U}_{\hat\omega} \subset \mathcal{I}_r(\mathbb{K}^d)$ to be the set of all $r$-tuples $\mathcal{A}=(A_1,\ldots,A_r)$ for which there exists an extremal norm $|\cdot|_{\mathcal{A}}$ satisfying

$$\|A_{\omega_n}\cdots A_{\omega_1}\|_{\mathcal{A}}<\varrho(\mathcal{A})^n$$

whenever $\hat\omega \nsim (\omega_1,\ldots,\omega_n)$. Openness conjecture. The set $\mathcal{U}_{\hat\omega}$ is an open subset of $\mathcal{I}_r(\mathbb{K}^d)$. This proposed generalisation would make a condition involving extremal norms and characteristic words more broadly useful, but the supplied source gives no resolution.