Neumann subspectrality for finite covers by normal manifolds

Let $M$ be a normal manifold, and let $\{\Gamma_i\}$ be a finite open cover of $M$ by normal manifolds. Give the interior boundary components Neumann conditions. The finite-cover subspectrality conjecture. The disjoint union

$$\bigsqcup_i\Gamma_i$$

is subspectral to $M$. This conjecture proposes a Neumann analogue of domain monotonicity for overlapping finite covers; the source does not state whether it is known or resolved.