Homological-lift conjecture with expected degree

Let $\mu$ be a stability condition as in Assumption (stability), let $\alpha\in C(\mathcal A)_\mathrm{pe}$, and let $\varepsilon_\alpha^\mu\in K_\ast(\mathfrak M^\mathrm{rig})$ be the associated class. A homological lift is denoted by $\varepsilon_\alpha^{\mu,H}$. Homological-lift and homogeneity conjecture. The class $\varepsilon_\alpha^\mu$ admits a homological lift with the expected homological degree

$$\varepsilon_\alpha^{\mu,H}\in H_{2-2\chi(\alpha,\alpha)}(\mathfrak M^\mathrm{rig}).$$

This is the precise expected-degree form of the preceding claim about canonical homological lifts without a framing functor; the supplied text gives no resolution.