Extension of the main symmetric-group-algebra theorem to Hecke shuffles

Let $\mathcal{H}=\mathcal{H}_q(S_n)$ be the type-A Iwahori–Hecke algebra with basis $(T_w)_{w\in S_n}$. For $1\leq \ell\leq n$, define the $q$-deformed somewhere-to-below shuffle by

$$t_{\ell}^{\mathcal{H}}:=T_{\operatorname{cyc}_{\ell}}+T_{\operatorname{cyc}_{\ell,\ell+1}}+T_{\operatorname{cyc}_{\ell,\ell+1,\ell+2}}+\cdots+T_{\operatorname{cyc}_{\ell,\ell+1,\ldots,n}}\in\mathcal{H}.$$

Hecke-algebra extension conjecture. Theorem should hold in $\mathcal{H}$ when each $t_\ell$ is replaced by $t_\ell^{\mathcal{H}}$.

This proposes that the main result for the original somewhere-to-below shuffles persists under the Hecke-algebra deformation. The source gives no proof or resolution, and the precise content of the referenced theorem is not included in the supplied context.