Failure of Boolean-ultrapower saturation invariance without distributivity

Let $cappa$ be a cardinal, let $cB$ be a complete Boolean algebra, and let $U$ be a $cappa$-regular ultrafilter on $cB$. Let $cM$ and $cN$ be elementarily equivalent $L$-structures with $|L|\leqcappa$.

Boolean-ultrapower saturation conjecture. There exist such $cappa$, $cB$, $U$, $cM$, and $cN$ for which

$$cM^{[cB]}/U\text{ is }cappa^+\text{-saturated},$$

while

$$cN^{[cB]}/U\text{ is not }cappa^+\text{-saturated}.$$

This conjectures that the saturation equivalence established for $\langlecappa,2\rangle$-distributive complete Boolean algebras can fail when the distributivity assumption is removed.