The sliding-relations conjecture for the Kauffman bracket skein module of two handlebodies

Let $H_n$# $H_m$ denote the connected sum of handlebodies of genera $n$ and $m$, let $\mathcal{S}_{2,\infty}(H_2)$ be the Kauffman bracket skein module of the genus-two handlebody, and let $\mathcal{I}$ be the ideal generated by the sliding relations. Sliding-relations conjecture. The Kauffman bracket skein module of the connected sum of two genus-one handlebodies satisfies

$$\mathcal{S}_{2,\infty}(H_1 \ \# \ H_1) = \mathcal{S}_{2,\infty}(H_2)/\mathcal{I}.$$

The paper has shown that the analogous general statement does not hold in full generality, but the calculations suggest that the sliding relations generate the relevant ideal in the case of $H_1 \ \# \ H_1$; the assertion is presented as a future direction and remains open here.