The stable ALE desingularization conjecture for spherical and hyperbolic orbifolds

Let $(N^4,g_b)$ be a stable Ricci-flat ALE $4$-manifold with tangent cone at infinity $\mathbb{R}^4/\Gamma$. Let $(M_o,g_o)$ be a spherical, respectively hyperbolic, orbifold with a singularity modeled on $\mathbb{R}^4/\Gamma$. Denote the connected sum along the link $\mathbb{S}^3/\Gamma$ by $M_o\#_{\mathbb{S}^3/\Gamma}N$.

Stable ALE desingularization conjecture. There exists an ancient, respectively immortal, renormalized Ricci flow on $M_o\#_{\mathbb{S}^3/\Gamma}N$ with limit $(M_o,g_o)$ at $-\infty$, respectively $+\infty$.

This extends the paper's constructions from the particular ALE spaces treated there to any stable Ricci-flat ALE $4$-manifold and relates the time direction to the stability type of the orbifold soliton.