The basis conjecture for the handlebody Hecke algebra

Let $H_{g,n}(q)$ be the Hecke algebra associated with the handlebody braid group, let $G_{g,n}$ be the corresponding group, and let $\Sigma_{g,n}$ be the set of elements

$$u_1u_2\ldots u_n\sigma,$$

where each $u_i$ is a reduced word in the free group

$$\langle t_{1,g+i},t_{2,g+i},\ldots,t_{g,g+i}\rangle\cong F_g,$$

and $\sigma$ is a basis element of $H_n(q)$. Basis conjecture. There is an isomorphism

$$H_{g,n}(q)\cong\mathbb{C}[G_{g,n}]$$

as $\mathbb{C}$-modules. In particular, $\Sigma_{g,n}$ is a basis of the algebra $H_{g,n}(q)$. This asserts that the normal-form basis coming from the handlebody braid group extends to the corresponding Hecke algebra; the supplied source does not indicate whether the claim has been resolved.