Unitary-equivalence conjecture for spectral monotonicity under geometric flows

Let $S$ be a Schrödinger operator evolving with a geometric flow $\mathcal{F}$, and suppose that its eigenvalues are non-decreasing under $\mathcal{F}$. Let $T$ be an operator unitarily equivalent to $S$. Unitary-equivalence conjecture. Then the eigenvalues of $T$ are also non-decreasing under $\mathcal{F}$. This follows naturally from the fact that unitarily equivalent operators have the same spectrum, but the source states it as a conjectural expectation and provides no separate resolution.