Faithfulness conjecture for the polynomial representation at slope one-half

Let $\alpha\in({\mathbb Z}^2)^+$, and let $A^{\frac12}_\alpha$ be the algebra constructed from the convolution homology of the corresponding moduli spaces. Its polynomial representation is the representation

$$P^{\frac12}_\alpha=\bigoplus_{\underline{\alpha}\in\mathrm{Seq}^{\frac12}(\alpha)}\bigotimes_i \mathrm{H}^*(\operatorname{Coh}^{\frac12}_{\alpha_i})$$

obtained from the convolution action.

Faithfulness conjecture. For any $\alpha\in({\mathbb Z}^2)^+$, the polynomial representation $P^{\frac12}_\alpha$ of $A^{\frac12}_\alpha$ is faithful.

For $\nu\neq\frac12$, faithfulness follows from a result of Przezdziecki, while the slope-$\frac12$ case is the remaining case proposed here.