Christ's Higgs–cosingularity equivalence conjecture with Hom-bijections

Let $\Gamma$ be the relative Ginzburg algebra, let ${\mathcal C}$ be its cluster category, let ${\mathcal H}\subseteq {\mathcal C}$ be the Higgs category, and let

$$\Phi:{\mathcal C}\longrightarrow {\rm cosg}(\Gamma)$$

be the quotient functor. For objects $X,Y\in {\mathcal H}$ and integers $n\geq 0$, write $\Sigma^{-n}Y$ for the corresponding negative shift.

Christ's conjecture. The restriction of $\Phi$ to ${\mathcal H}$ is an equivalence of $k$-linear categories

$${\mathcal H}\xrightarrow{\sim}{\rm cosg}(\Gamma),$$

and it induces bijections

$${\rm Hom}_{\mathcal C}(X,\Sigma^{-n}Y)\xrightarrow{\sim}{\rm Hom}_{{\rm cosg}(\Gamma)}(X,\Sigma^{-n}Y)$$

for all $X,Y\in {\mathcal H}$ and all integers $n\geq 0$. This gives a precise form of the expected equivalence between the Higgs and cosingularity categories, strengthening it by requiring compatibility with all non-positive shifts. The supplied source gives no evidence of resolution.