Commutator nilpotency for somewhere-to-below shuffles

Given a positive integer $n$, we consider the group algebra of the symmetric group $S_n$. In this algebra, we define $n$ elements $t_1,t_2,\dots ,t_n$ by the formula $$t_{\mathrm{\ell }}:=\underset{\mathrm{\ell }}{\mathrm{cyc}}+\underset{\mathrm{\ell },\mathrm{\ell }+1}{\mathrm{cyc}}+\underset{\mathrm{\ell },\mathrm{\ell }+1,\mathrm{\ell }+2}{\mathrm{cyc}}+\cdots +\underset{\mathrm{\ell },\mathrm{\ell }+1,\dots ,n}{\mathrm{cyc}},$$ where ${\mathrm{cyc}}_{\mathrm{\ell },\mathrm{\ell }+1,\dots ,k}$ denotes the cycle that sends $\mathrm{\ell }\mapsto \mathrm{\ell }+1\mapsto \mathrm{\ell }+2\mapsto \cdots \mapsto k\mapsto \mathrm{\ell }$. These $n$ elements are called the somewhere-to-below shuffles due to an interpretation as card-shuffling operators. In this paper, we show that their commutators $[t_i,t_j]=t_i{t}_j-t_j{t}_i$ are nilpotent, and specifically that $${[t_i,t_j]}^{\lceil (n-j)\mathrm{/}2\rceil +1}=0\text{ for any }i,j\in \{1,2,\dots ,n\}$$ and $${[t_i,t_j]}^{j-i+1}=0\text{ for any }1\le i\le j\le n\mathrm{.}$$ We discuss some further identities and open questions.