Generic character-reduced skein-module irreducibility and dimension conjecture for handlebodies

Let $H_g$ be the genus $g$ handlebody, let $\rho\in \mathfrak{X}_{{\rm SL}_3(\mathbb{C})}(H_g)$ be represented by an irreducible representation, and let $\mathscr S_{\bar\omega}(H_g)_\rho$ be the character-reduced skein module. Handlebody representation conjecture. The vector space $\mathscr S_{\bar\omega}(H_g)_\rho$ is an irreducible representation of $\mathscr S_{\bar\omega}(\partial H_g)$ with dimension $N^{8g-8}$ whose classical shadow is $\rho_\partial$. This predicts irreducibility and the expected dimension for the representation arising from an irreducible character; the source gives no resolution status.