Gouvêa–Mazur spectral expansion conjecture, strong form

Let $p$ be prime, let $r\in(\frac{1}{p+1},\frac{p}{p+1})$, and let $h$ be an $r$-overconvergent modular function on the region $X_0(1)_{\ge p^{-r}}$. Its spectral expansion is the expansion in the eigenfunctions of the compact operator $U$. Gouvêa–Mazur spectral expansion conjecture. The spectral expansion of $h$ converges to $h$ in the supremum norm of $X_0(1)_{\ge p^{-r}}$. This conjecture would establish norm-convergent spectral expansions for all overconvergent modular functions in the stated range; the source presents it as an optimistic conjecture, and no resolution is supplied.