The boundedness conjecture for smooth and proper stable categories

Let $\mathcal{C}$ be a stable $dg$-category over a field $k$, equipped with a stability condition $\sigma$. Say that $\sigma$ has a mass-Hom bound if for every $E \in \mathcal{C}$ there is a constant $C_E$ such that

$$\dim_k \operatorname{Hom}(E,F) \leq C_E m_\sigma(F), \qquad \forall F \in \mathcal{C}.$$

Boundedness conjecture. If $\mathcal{C}$ is smooth and proper over $k$, then every stability condition on $\mathcal{C}$ admits a mass-Hom bound.

This conjecture is presented as a consequence of the manifold-with-corners conjecture in the paper. The supplied text does not give an independent resolution of it.