In their 1972 paper [On the number of complete Boolean algebras](https://link.springer.com/article/10.1007/BF02945048) Monk and Solovay showed that if $\lambda$ is an infinite cardinal, then there are $2^{2^\lambda}$ many  isomorphism types of
complete Boolean algebras of power $2^\lambda.$ 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?