Interlacing conjecture for Neumann and Dirichlet eigenvalues of self-similar measures

Let $\mu$ be a self-similar measure with contraction parameters $r_1,r_2$ and weights $m_1,m_2$ satisfying

$$r_1m_1=r_2m_2.$$

Write $\lambda_{N,k}$ and $\lambda_{D,k}$ for the Neumann and Dirichlet eigenvalues, respectively. Interlacing conjecture. The eigenvalues satisfy

$$\lambda_{N,0}<\lambda_{N,1}<\lambda_{D,1}<\lambda_{D,2}<\lambda_{N,2}<\lambda_{N,3}<\lambda_{D,3}<\lambda_{D,4}<.\,.\,.\,.$$

This conjecture is motivated by the examination of several examples and proposes a specific ordering of the Neumann and Dirichlet spectra for self-similar measures under the balance condition $r_1m_1=r_2m_2$; its resolution is not established in the supplied text.