RCD no-\texorpdfstring{$\mathbb{RP}^2$}{RP2} cone conjecture

Fix $n\ge 4$. Let $(X_i,\mathsf{d}_i,\mathscr{H}^n,p_i)$ be $\operatorname{RCD}(-\delta_i,n)$ spaces, with $\delta_i\to 0$, $\mathscr{H}^n(B_1(p_i))>v>0$, and suppose that

$$(X_i,\mathsf{d}_i,p_i)\xrightarrow{\mathrm{pGH}}\mathbb{R}^{n-3}\times C(Y)\,,\quad i\to\infty\,,$$

for some metric space $(Y,\mathsf{d}_Y)$. Assume that each $(X_i,\mathsf{d}_i)$ has no blow-up of the form $\mathbb{R}^{n-3}\times C(W)$ with $W\approx\mathbb{RP}^2$.

RCD no-$\mathbb{RP}^2$ cone conjecture. Then $Y\approx S^2$ or $Y\approx \overline{D}^2$. If the $X_i$ have empty boundaries, only the first possibility can occur.

The conjecture seeks to generalize a stability result for singularities of the form $\mathbb{R}^{n-3}\times C(\mathbb{RP}^2)$ from smooth spaces to RCD spaces. The source identifies the lack of a suitable RCD slicing argument and the essential use of smoothness in the existing proof as major obstacles; its resolution status is not specified.