The eigenfunction-space conjecture for spectral projections on Sierpinski fractafolds

Let $\Gamma$ be the graph under consideration, let $\mu$ be the spectral measure, and let $P_\lambda$ and $\tilde{P_\lambda}$ denote the spectral projection kernels introduced above. For $\mu$-almost every $\lambda$, write $\xi_\lambda$ for a Hilbert space of $\lambda$-eigenfunctions with inner product $\langle\,\ \rangle_\lambda$. Eigenfunction-space conjecture. For every $f\in\ell^2(\Gamma)$, one has $P_\lambda f\in\xi_\lambda$ for $\mu$-almost every $\lambda$, and

$$\langle P_\lambda f,f\rangle=\langle P_\lambda f,P_\lambda f\rangle_\lambda.$$

Moreover, an analogous statement holds with $\tilde{P_\lambda}F$ and $\langle\tilde{P_\lambda}F,F\rangle$. This would express the spectral densities in the Plancherel formula using inner products on spaces of $\lambda$-eigenfunctions; the source does not provide a resolution, so the conjecture remains open.