Convergence conjecture for the Temperley–Lieb algebra rewriting system

Let $n$ be the rank parameter of the presented Temperley–Lieb algebra, with generators $\delta,e_1,\dots,e_{n-1}$. Consider the rewriting system on words in $\{\delta,e_1,\dots,e_{n-1}\}$ whose rules are

$$e_i\delta\to\delta e_i,\qquad e_i^2\to\delta e_i,$$ $$e_i e_{i\pm1}e_i\to e_i,\qquad e_i e_j\to e_j e_i\text{ if }j<i-1.$$

Convergence conjecture. This rewriting system is convergent, meaning that it is terminating and confluent.

If true, every word has a unique normal form, yielding an algorithm for reducing words and thereby solving equality in the presented Temperley–Lieb algebra.