Quotient structure conjecture for the loop Hecke algebra

Let $L\!H_n$ be the loop Hecke algebra, let $\chi^{(j+1)}$ be the element defined by the product of the differences of the braid and symmetric generators, and let ${\mathcal M}^p_n$ denote the matrix used in the source. Consider the quotient $L\!H_n/\chi^{(j+1)}$.

Quotient structure conjecture. The structure of $L\!H_n/\chi^{(j+1)}$ is given by the $j\times j$ truncation of ${\mathcal M}^p_n$.

The source says that this assertion has been checked up to rank $5$, but supplies no proof for arbitrary rank.