Hecke-algebra analogue of the somewhere-to-below shuffle commutator bounds

Let $\mathcal{H}=\mathcal{H}_q(S_n)$ be the type-A Hecke algebra, with basis elements $T_w$ and $q$-deformed somewhere-to-below shuffles $t_\ell^{\mathcal{H}}$. The source defines these deformations by

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

Hecke deformation conjecture. Corollary 28 and Corollary 29 both seem to hold in $\mathcal{H}$ when every $t_\ell$ is replaced by $t_\ell^{\mathcal{H}}$.

This proposes that the commutator-nilpotency bounds established for the original somewhere-to-below shuffles persist under the type-A Hecke deformation. The statement is presented as an observed generalization, with no proof or resolution supplied in the source.