Spectral-mass asymptotics on the infinite Sierpinski gasket

Let $SG$ denote the Sierpinski gasket and $SG_{\infty}$ its infinite version, and let $M(\lambda)$ denote the spectral-mass function. Infinite-gasket asymptotics conjecture. The function $M(\lambda)$ is well-defined on $SG_{\infty}$ and has the same asymptotics as on $SG$:

$$M(\lambda)\text{ satisfies the asymptotics (7.3) for }SG.$$

The claim is motivated by an averaging heuristic for localized eigenfunctions on the infinite gasket. The source gives no proof or resolution.