The higher-order GJMS geometric Aubin set conjecture

Let $k,n\in\mathbb{N}$ with $n>2k$. For a compact conformal $n$-manifold $(M^n,\mathfrak{c})$, let $Y_{L_{2j}}(M,\mathfrak{c})$ denote the Yamabe constant associated with the conformally covariant operator $L_{2j}$, and define

$$\mathscr{A}_{n,2k}:=\left\{(M^n,\mathfrak{c}): Y_{L_2}(M,\mathfrak{c}),\dotsc,Y_{L_{2k}}(M,\mathfrak{c})>0\right\}.$$

The higher-order GJMS geometric Aubin set conjecture. The set $\mathscr{A}_{n,2k}$ is a geometric Aubin set for $L_{2k}$.

This conjecture generalizes the known conformal Laplacian and Paneitz examples. It is known under additional structural assumptions on the conformal manifold, but remains open in general; Mazumdar's criterion reduces it to the Strong Maximum Principle and strict comparison with the spherical Yamabe constant.