Kronheimer–Ricci-flat metric coincidence conjecture for the resolved threefold

Let $Y^\Gamma_{[3]}$ be the resolved threefold carrying a line-bundle structure, and let $\mathcal{ED}$ denote its exceptional divisor. Write $\mathrm{ds}^2_{Kro}[Y^\Gamma_{[3]}]$ for the Kronheimer Kähler metric and $\mathrm{ds}^2_{Ricflat}[Y^\Gamma_{[3]}]$ for a Ricci-flat metric on the same manifold, with the same isometries and asymptotically locally flat behavior.

Kronheimer–Ricci-flat coincidence conjecture. The two metrics are different on $Y^\Gamma_{[3]}$, but they coincide on the exceptional divisor $\mathcal{ED}$.

This is a more specific formulation of the metric comparison for the resolved threefold. The source notes that the Ricci-flat metric is more precisely quasi-ALE and attributes that terminology to Joyce; it gives no resolution status for the conjecture.