Freeness conjecture for morphism spaces in the diagrammatic q-Schur category

Let $\lambda\rightarrow\ru(n,d)$ be a weight and let $\mathbf i$ and $\mathbf j$ be sequences for which the displayed 1-morphisms are defined. The morphism space is $\operatorname{HOM}_{\mathcal{S}(n,d)}(\mathcal{E}_{\mathbf i}1_{\lambda},\mathcal{E}_{\mathbf j}1_{\lambda})$, and its coefficient algebra is $\operatorname{End}_{\mathcal{S}(n,d)}(1_{\lambda})$. Freeness conjecture. The right module

$$\operatorname{HOM}_{\mathcal{S}(n,d)}(\mathcal{E}_{\mathbf i}1_{\lambda},\mathcal{E}_{\mathbf j}1_{\lambda})$$

is free of finite rank over

$$\operatorname{End}_{\mathcal{S}(n,d)}(1_{\lambda}).$$

The preceding dot-reduction argument proves finite generation over the endomorphism algebra, while freeness is left conjectural. The source also states a corresponding left-module version when $\mathcal E_{\mathbf i}1_{\lambda}=1_{mu}\mathcal E_{\mathbf i}$ and similarly for $\mathbf j$; this is merged here as the same structural conjecture.