The spectral-polynomial identity for the Heun equation

Let $H$ be the Hamiltonian whose restriction to the finite-dimensional invariant space $V$ has monic characteristic polynomial $P(E)$, and let $Q(E)$ be the polynomial arising from the commuting-operator construction. The preceding proposition establishes that the zero sets of $P(E)$ and $Q(E)$ coincide.

Spectral-polynomial identity. The two polynomials are equal:

$$P(E)=Q(E).$$

This identity would strengthen equality of the zero sets to equality of the monic polynomials themselves. The source presents it as a conjecture; no resolution is supplied here.