HRS-tilt characterization conjecture for weighted projective lines

Let $cX$ be a weighted projective line of arbitrary type, and let $(cD^{\leq 0},cD^{\geq 0})$ be a bounded t-structure on $cD^b(cX)$. Let $(\mathcal{T},\mathcal{F})$ be a torsion pair in $\operatorname{coh}\mathbb{X}$, and let its HRS-tilt denote the associated tilted heart. A nonzero object of $cD^{\leq 0}$ is Ext-projective when it has no nontrivial positive Ext against the relevant objects.

HRS-tilt characterization conjecture. The aisle $cD^{\leq 0}$ contains no nonzero Ext-projective if and only if it is a shift of the HRS-tilt associated with some torsion pair $(\mathcal{T},\mathcal{F})$ in $\operatorname{coh}\mathbb{X}$ such that there is no nonzero sheaf $E\in\mathcal{T}$ with $\tau E\in\mathcal{F}$.

This is proposed as a potential approach to the arbitrary-type derived-equivalence conjecture. Its status is not resolved in the supplied text.