Is it consistent with, or even implied by, CH that there is a CH-preserving, powerfully ccc complete Boolean algebra $\mathbb{B}$ of size $2^{\aleph_1}$? (Powerfully ccc means that ccc holds in every finite power of $\mathbb{B},$ or equivalently in every finite support power of $\mathbb{B}$).
As Monroe discusses here, a CH-preserving ccc forcing $\mathbb{B}$ factors as $\mathbb{A} * \dot{\mathbb{S}},$ where $\mathbb{A}$ has cardinality at most $\aleph_1$ and $\dot{\mathbb{S}}$ is a name for a Suslin algebra (ie a ccc $\omega$-Baire algebra), the latter having cardinality at most $2^{\aleph_1}.$ Every nontrivial Suslin algebra has a Suslin tree as a subalgebra, so its square is not ccc. Therefore $\mathbb{A}$ has to be nontrivial.
Although this is a ZFC problem, the need for such a forcing has come up in my choiceless work, regarding how to blow up $\aleph(\mathcal{P}(X))$ while keeping $\aleph^*(\mathcal{P}_{\text{bnd}}(\aleph^*(X)))$ small.
