Conjecture on complete fractional Yamabe metrics with high-dimensional singularities

Let $n\geq 3$, $s\in(0,1)$, and let $k\in((n+2s)/2,n)$ satisfy, for some $j=0,1,\dots$, either

$$\frac{n+2s}{2}+4j<k<\frac{n-2s}{2}+2+4j,$$

or

$$\frac{n+2s}{2}+2+4j<k\leq\frac{n-2s}{2}+4+4j.$$

Let $\Sigma^k$ be a smooth compact $k$-dimensional submanifold without boundary in $\mathbb{R}^n$.

High-dimensional fractional Yamabe singularity conjecture. There exists a complete Yamabe metric that is singular on $\Sigma^k$.

This conjecture addresses non-distributional solutions with singularities of dimension above the distributional maximality threshold $(n-2s)/2$. The source presents these parameter ranges as conjectural, and no resolution is given in the supplied text.