Conjecture on approximate joint eigenstates from the Clifford linear pseudospectrum

Let $\bm{A}=(A_1,\ldots,A_d)$ be a tuple of Hermitian operators, let $B$ be a non-Hermitian matrix, and let $\bar{\Lambda}_{\epsilon}^{\mathrm{C}}(\bm{A},B)$ denote the Clifford linear $\epsilon$-pseudospectrum. Let $(\bm{\lambda},\nu)$ be a probe site in this pseudospectrum, with $\bm{\lambda}=(x_1,\ldots,x_d)$ and $\nu=E$. A unit state is a vector $\bm{\psi}$ with $\lVert\bm{\psi}\rVert=1$.

Approximate joint-eigenstate conjecture. If $(\bm{\lambda},\nu)\in\bar{\Lambda}_{\epsilon}^{\mathrm{C}}(\bm{A},B)$, then there is a unit state $\bm{\psi}$ such that

$$X_i\bm{\psi}\approx x_i\bm{\psi},\qquad H\bm{\psi}\approx E\bm{\psi}.$$

This conjecture proposes an approximate common eigenstate interpretation for points in the Clifford linear pseudospectrum. The source notes that the analogous result and proof are not clear; the notation $X_i$, $H$, $x_i$, and $E$ is not defined in the supplied context, so the precise intended identification with $A_i$, $B$, $\lambda_i$, and $\nu$ should be checked.