The handlebody skein-representation composition-factor conjecture

Let $\Sigma_g$ be a closed genus-$g$ surface and $H_g$ a genus-$g$ handlebody. The reduced skein module $\mathcal{S}_q^{\mathrm{red}}(H_g)$ carries the representation of the skein algebra $\mathcal{S}_q(\Sigma_g)$ induced by its action on the handlebody. Let $J_i$ denote the terms in a composition series of the representation induced by the Wilson loop map

$$W:\mathcal{S}_q(\Sigma_g)\longrightarrow\mathcal{L}_{g,0}(\bar U_q)$$

and the representation of $\mathcal{L}_{g,0}(\bar U_q)$ on $(\bar U_q^*)^{\otimes g}$. Handlebody skein-representation composition-factor conjecture. The representation of $\mathcal{S}_q(\Sigma_g)$ on $\mathcal{S}_q^{\mathrm{red}}(H_g)$ is a composition factor $J_{i+1}/J_i$ of that representation. In genus one this composition-factor relationship is established, while the conjecture proposes its persistence in every genus; the cited irreducibility result for the handlebody skein representation motivates the claim.