The generic nonvanishing conjecture for conformally invariant nonlinear Dirac equations

Let $M$ be a connected Riemannian spin manifold, and set

$$r:=\frac{2}{n-1}.$$

Let $\phi$ be a nonzero spinor satisfying

$$D^{g_0}\phi=c|\phi|^r\phi$$

for some Riemannian metric $g_0$ and $c\in\mathbb{R}$. Generic nonvanishing conjecture. For generic conformal classes on $M$, every such solution is everywhere non-zero. The exponent is special because the nonlinear Dirac equation and the zero set of $\phi$ are conformally invariant; the claim is proposed as a generalization of the generic nonvanishing result known in the compact $2$- and $3$-dimensional eigenspinor setting.