Decidability of Boolean algebras with ideal quantification

Let $\lambda$ be a cardinal satisfying

$$\lambda\leq 2^{\aleph_0}.$$

A Boolean algebra has ideal quantification when its first-order language is expanded by allowing quantification over ideals.

Boolean-algebra decidability conjecture. The theory of Boolean algebras of cardinality less than $\lambda$, or the first-order theory of Boolean algebras expanded by quantification over ideals, is decidable when

$$\lambda\leq 2^{\aleph_0},$$

including the case $\lambda=\aleph_2\leq 2^{\aleph_0}$.

The parser supplies no resolution for this candidate. The surrounding text relates it to decidability results for countable Boolean algebras and linear orders, but does not establish the stated cardinal-general assertion.