Class numbers in cyclotomic ZpZp-extensions

For any prime p and any positive integer n  , let Bp,nBp,n denote the n  th layer of the cyclotomic ZpZp-extension over the rationals. Based on the heuristics of Cohen and Lenstra, and refined by new results on class numbers of particular Bp,nBp,n, we provide evidence for the following conjecture: For all primes p and positive integers n  , the ideal class group of Bp,nBp,n is trivial.