Conjecture on point spectrum of self-adjoint extensions via Weyl relations

Let the assumptions of hold, with $\alpha$ the point under consideration, $m$ and $\tau$ the corresponding relations, and $\widetilde A$ the self-adjoint extension. For a relation $r$, write $r(\alpha)$ for its value at $\alpha$, and call an element nontrivial when it is nonzero.

Point-spectrum conjecture. One has $\alpha\in\sigma_p(\widetilde A)$ if and only if either there exists a nontrivial element $y_0$ such that

$$(0;y_0)\in m(\alpha)\cap\tau(\alpha),$$

or there exists a nontrivial element $x_0$ such that

$$(x_0;0)\in (m+\tau)(\alpha).$$

The conjecture proposes that, when $m$ and $\tau$ do not share a pole vector at $\alpha$, the cancellation effects seen in earlier counterexamples cannot occur. It characterizes eigenvalues of the self-adjoint extension through common vertical elements of $m(\alpha)$ and $\tau(\alpha)$ or horizontal elements of their sum; the supplied text gives no resolution, so the status remains open.