Homological modification of mutation for flips

Work in the setting of Cohen–Macaulay rational singularities. Let $N=H^0(\mathcal{V}_X)$, where $\mathcal{V}_X$ is the object from the paper’s contraction and tilting setup, and let $X'$ be the flip of the relevant contraction. Homological modification conjecture for flips. There is a homological modification of mutation that produces $H^0(\mathcal{V}_{X'})$ from $N$; consequently, the Homological MMP can be extended to cover both flips and flops. The proposed extension would distinguish flips from flops by modifying the algebra rather than relying on derived-equivalence-based stability tracking; it remains open.