Bijectivity of the silting-to-cluster-tilting map

Let $({\mathcal T},{\mathcal T}^{\mathsf{fd}},{\mathcal M})$ be the $(d+1)$-Calabi--Yau triple from Corollary~(b), let ${\mathcal C}$ be the associated triangulated category, and let ${\mathcal F}$ be the fundamental domain used there. Write $\operatorname{silt}\nolimits^{\mathcal F}{\mathcal T}$ for the silting subcategories of ${\mathcal T}$ contained in ${\mathcal F}$, and write ${\it d}\strut\kern-.2em\operatorname{-ctilt}\nolimits{\mathcal C}$ for the $d$-cluster-tilting subcategories of ${\mathcal C}$. The functor $\pi\colon {\mathcal T}\to{\mathcal C}$ induces a map

$$\pi\colon\operatorname{silt}\nolimits^{\mathcal F}{\mathcal T}\to{\it d}\strut\kern-.2em\operatorname{-ctilt}\nolimits{\mathcal C}.$$

Bijectivity conjecture. The map $\pi\colon\operatorname{silt}\nolimits^{\mathcal F}{\mathcal T}\to{\it d}\strut\kern-.2em\operatorname{-ctilt}\nolimits{\mathcal C}$ is bijective for all $d\geq 1$.

The map is known to be bijective for $d=1$ and $d=2$; the conjecture asks whether this remains true for all higher values of $d$.