cscK blowup conjecture with moment-map perturbation

Let $(M,L)$ be a polarized compact complex manifold of dimension $m$, with a constant-scalar-curvature Kähler metric in $c_1(L)$, and let $p\in M$. Let $\mu$ be the moment map and let $\Delta\mu$ denote its Laplacian; let $E$ be the exceptional divisor of the blowup $\pi:Bl_pM\to M$.

cscK blowup conjecture with moment-map perturbation. If $m>2$, there exist $\delta_0,\varepsilon_0>0$ such that, whenever $\mu(p)+\delta\Delta\mu(p)=0$ for some $\delta\in(0,\delta_0)$, then for every $\varepsilon\in(0,\varepsilon_0)$ the manifold $Bl_pM$ admits a cscK metric in the Kähler class

$$c_1(\pi^*L-\varepsilon E).$$

If $m=2$, the analogous statement should hold using $\mu(p)\pm\delta\Delta\mu(p)$, with the sign determined by the cases in the preceding stability theorem.

This is proposed as a strengthening of the paper’s main blowup theorem and is intended to establish sharpness of the stated stability result, at least when $m>2$. The source gives no resolution or subsequent status for this conjectural strengthening.