Conjecture on continuity of the Clifford linear gap function

Let $\bm{A}=(A_1,\ldots,A_d)$ be a tuple of Hermitian operators, let $B$ be a non-Hermitian matrix, and let $(\bm{\lambda},\nu)\in\mathbb{R}^d\oplus\mathbb{C}$ be a probe site. The Clifford linear gap function is

$$\bar{\mu}_{(\bm{\lambda},\nu)}^{\mathrm{C}}(\bm{A},B)=\min\left\lvert\operatorname{Re}\left(\operatorname{Spec}\left(L_{(\bm{\lambda},\nu)}(\bm{A},B)\right)\right)\right\rvert.$$

Continuity conjecture. The Clifford linear gap function is pointwise continuous.

The conjecture concerns the stability of the real-part spectral gap of the non-Hermitian spectral localizer as the probe site and operator data vary. The source gives numerical evidence but no resolution, so the conjecture remains open.