Free-module basis conjecture for the principal right ideal

Let $R=\mathbb{F}[x_1,\ldots,x_d]$, let $Te_0^*$ be the principal right ideal, and call a $ZZ$ word feasible when it is nontrivial, ends in $e_0^*$, and has mutually distinct indices. Free-module basis conjecture. The $R$-module $Te_0^*$ is free of rank $2^d$, with an $R$-basis consisting of the feasible $ZZ$ words. This conjecture predicts a concrete free-module structure and basis for $Te_0^*$; it is proposed for future research and is not proved in the paper.