Typicality of non-equidistribution for bounded p-divisibility

Determine whether, for curves over discretely valued fields and sequences of primes with bounded p-divisibility, the non-equidistribution behavior exhibited by the constructed supersingular examples is typical.

Background

The paper proves equidistribution for sequences of modular curves $Y_0(a_n)$ whose levels have $p$-adic valuations tending to infinity. It also constructs a family of sequences of primes $a_n\neq p$, for which the levels have bounded $p$-divisibility, and shows that equidistribution fails because a positive proportion of intersection points remains concentrated near a supersingular point.

After presenting this counterexample, the paper states that the observed failure of equidistribution is believed to be typical for curves defined over discretely valued fields and sequences of levels with bounded $p$-divisibility. Establishing or disproving that typicality is therefore left unresolved.