Optimality of a delta distribution for the lowest Dirac-Coulomb eigenvalue

Let $D_0=-i\boldsymbol{\alpha}\cdot\nabla+\beta$ be the free Dirac operator, and let $\mu$ be a non-negative finite measure on $\mathbb{R}^3$ with Coulomb potential $V_\mu=\mu\ast|x|^{-1}$. Write $\lambda_1(D_0-V_\mu)$ for the lowest eigenvalue, and let $\lambda_1(\nu)$ denote the corresponding value when $\mu=\nu\delta_0$.

Optimality for a delta. The first eigenvalue is minimal when the charge distribution is concentrated at the origin, namely when $\mu=\mu(\mathbb{R}^3)\delta_0$. Equivalently, for all $0\leqslant\nu<1$,

$$\lambda_1(\nu)=\lambda_1(D_0-\nu|x|^{-1})=\sqrt{1-\nu^2}.$$

In particular, this would imply

$$\nu_0=\nu_1=1.$$

For the Schrödinger operator, the analogous delta-optimality statement is well known. The conjecture asks whether the same phenomenon holds for the Dirac operator and would determine the critical constants $\nu_0$ and $\nu_1$.