Faithfulness of the heart-group functor for finite-dimensional algebras

Let $\Lambda$ be a finite-dimensional algebra. The group $\operatorname{\mathrm{int}\text{-}\mathrm{heart}}(\operatorname{tors}\Lambda)$ has heart-controlled generators, meaning that for closed intervals $[\mathcal{U},\mathcal{T}]$ and $[\mathcal{U}',\mathcal{T}']$ in $\operatorname{tors}\Lambda$, equality of the corresponding generators is equivalent to $\mathcal{U}^{\perp}\cap\mathcal{T}=(\mathcal{U}')^{\perp}\cap\mathcal{T}'$. The conjecture. The functor $\Gamma$ from Theorem~ is a faithful group functor

$$\mathfrak{W}(\Lambda)\longrightarrow \operatorname{\mathrm{int}\text{-}\mathrm{heart}}(\operatorname{f-tors}\Lambda).$$

By Theorem~ and Proposition~, faithfulness of $\Gamma$ follows from the heart-controlled-generator property. The conjecture asserts that this property holds for every finite-dimensional algebra.