A basis conjecture for the Partition Temperley–Lieb algebra

Let ${\rm PTL}_n$ be the Partition Temperley–Lieb algebra, let $\mathsf{P}_n$ denote the set-partition monoid on $\{1,\ldots,n\}$, and let $F$ be a reduced word in the generators used for ${\rm PTL}_n$. Basis conjecture. The elements

$$E_I F,\qquad I\in\mathsf{P}_n,$$

form a linear basis of ${\rm PTL}_n$. Consequently,

$$\dim({\rm PTL}_n)=b_n C_n.$$

The conjecture concerns the spanning set constructed from the presentation of the Partition Temperley–Lieb algebra; proving linear independence would determine its dimension.