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