$L^p$ smallness conjecture for three-dimensional Dirac operators

Let $n=3$, let $V$ be a potential, and let $d4a0_{0,V}$ and $d4a0_{m,V}$ denote the massless and massive Dirac operators with potential $V$, respectively. Let spectral stability mean that the conclusions denoted by the source as the spectral relations for $d4a0_{0,V}$ and $d4a0_{m,V}$ hold.

$L^p$ smallness conjecture. There exists a positive constant $d4c$ independent of $V$ such that

$$\|V\|_{L^3}<d4c$$

implies spectral stability for $d4a0_{0,V}$, while

$$\|V\|_{L^3}+\|V\|_{L^{3/2}}<d4c$$

implies spectral stability for $d4a0_{m,V}$.

This is a concrete proposed integral replacement for the pointwise assumptions in the Dirac spectral-stability theorem. The source gives no resolution, so the claim remains open.