In their 1972 paper On the number of complete Boolean algebras Monk and Solovay showed that if $\kappa$ is an infinite cardinal, then there are $2^{2^\kappa}$ many isomorphism types of complete Boolean algebras of power $2^\kappa.$ On the other hand, by a result of Pierce, a complete Boolean algebra of infinite power $\kappa$ exists iff $\kappa=\kappa^{\aleph_0}$.

In the above mentioned paper, the following is asked:

Question. Suppose $\kappa=\kappa^{\aleph_0}$, but it is not of the form $2^\lambda$ for any $\lambda$. Are there ${2^\kappa}$ many isomorphism types of complete Boolean algebras of power $\kappa?$

What is known about the above question?