Thanks! But that one I found already, and I think it's a different vertical direction about powers of a single prime p rather than allowing p to vary over all primes.

@TimothyChow Yes, in fact the article is published. I recall it getting rejected from a journal at first, because the results were more negative than positive. But then it ended up published by Cambridge Univ. Press in the volume "Modular Forms on Schiermonnikoog", in 2008.