Does multifractality of electron states raise $T_{c}$ in disordered superconductors?
In plain words
Near the point where disorder traps electrons, their wavefunctions become patchy at every length scale (multifractal) and overlap strongly in hot spots, which theory predicts can strengthen pairing and raise the superconducting transition temperature. Whether this happens in real materials, and by how much, is not settled.
Precise statement
For disordered films with short-range attraction, theory (Feigel'man, Ioffe, Kravtsov and Yuzbashyan, PRL 2007; Burmistrov, Gornyi and Mirlin, PRL 2012) predicts $T_c$ increasing with disorder near the Anderson transition through multifractal eigenfunction correlations, with a spatially inhomogeneous gap of roughly log-normal distribution, competing with Coulomb suppression of $T_c$. Determine whether $T_c(\mathrm{disorder})$ rises in a material class where the theory applies, separate the multifractal effect from density-of-states and phonon changes, and give its size relative to the Coulomb suppression.
What would settle it
$T_c$, tunneling-gap statistics and local density-of-states correlations measured across a controlled disorder series, compared quantitatively with the multifractal theory including Coulomb repulsion.
Status in the literature
Scanning tunneling studies of monolayer niobium dichalcogenides (Zhao et al., Nature Physics 15, 904, 2019; Rubio-Verdu et al., arXiv 1810.08222) reported multifractal spatial structure of the superconducting gap; whether and by how much this raises $T_{c}$ against Coulomb suppression is unresolved.