Faithfulness conjecture for the action on the principal right ideal

Let $R=\mathbb{F}[x_1,\ldots,x_d]$, let $Te_0^*$ be the principal right ideal with its right $R$-module structure induced by the map $\mu$, and let $\operatorname{End}_R(Te_0^*)$ be the algebra of $R$-linear endomorphisms. The left action of $T$ gives an algebra homomorphism $T\to\operatorname{End}_R(Te_0^*)$. Faithfulness conjecture. The map $T\to\operatorname{End}_R(Te_0^*)$ is injective. This asserts that the action of $T$ on $Te_0^*$ is faithful; it is stated as a future-research conjecture and remains open in the source.