Is the primordial curvature spectrum enhanced on small scales?
In plain words
The microwave background shows that early density ripples had a strength of about one part in 20,000 on large scales. Forming black holes needs ripples thousands of times stronger on much smaller scales, which some inflation models predict.
Precise statement
The curvature power spectrum is $P_{\zeta} \sim 2.1e-9$ at $k = 0.05\,\mathrm{Mpc}^{-1}$, while a cosmologically relevant PBH abundance needs $P_{\zeta} \sim 1e-2$ at the scale entering the horizon at formation. Measure or bound $P_{\zeta}(k)$ for $k > 1\,\mathrm{Mpc}^{-1}$ using CMB spectral distortions (the COBE/FIRAS mu-distortion limit is about three orders of magnitude stronger than the PBH limit at $k \sim 1e1 \text{ to } 1e4\,\mathrm{Mpc}^{-1}$, Chluba, Erickcek and Ben-Dayan 2012), scalar-induced gravitational waves in pulsar-timing, LISA and ground-based bands, and ultracompact minihalos. An answer is a detected enhancement or an upper bound below the PBH-forming level at each $k$.
What would settle it
A PIXIE-class spectral-distortion measurement plus scalar-induced gravitational-wave searches by pulsar timing arrays and LISA covering the PBH-relevant k range.