Multiplicity invariance under left-definite theory

Let ${\bf A}$ be a semi-bounded self-adjoint operator in a Hilbert space $\mathcal{H}$, and let $\lambda$ be an eigenvalue of ${\bf A}$ with multiplicity $m$. Let ${\bf A}_r$ be the $r$-th left-definite operator in the $r$-th left-definite space $\mathcal{H}_r$. Multiplicity invariance conjecture. The number $\lambda$ is also an eigenvalue of ${\bf A}_r$ with multiplicity $m$. The source presents this as a related conjecture motivated by spectral stability; no resolution is given.