The injectivity conjecture for the homomorphism from $\tilde T$ to $\mathcal A\otimes T$

Let $\tilde T$ and $T$ be the algebras defined in the paper, and let $\mathcal A$ be the algebra acting on finite-dimensional irreducible modules described in the surrounding results. Let $\Phi:\tilde T\to\mathcal A\otimes T$ be the homomorphism sending $e_i$ to $(1-e)\otimes E_i$ and $e_i^*$ to $(1-f)\otimes E_i^*$ for $0\leq i\leq d$. Injectivity conjecture. The map $\Phi$ is injective. The claim concerns whether the constructed homomorphism embeds $\tilde T$ into $\mathcal A\otimes T$; the supplied text does not indicate whether it has been resolved.