Consistency of the Boolean-algebra splitting property

Let $D_{\omega_2}$ be the club filter on $\omega_2$, and let $F$ be the function used in the source to associate to a subset of $\omega_2$ the relevant set of ordinals. The property $(*)$ asserts that for every stationary

$$A\subseteq\{\alpha<\omega_2:{\rm cf}(\alpha)=\omega\}$$

and suitable $B,C$ with $F(A)=B\cup C$, there are stationary disjoint $A',B'$ such that

$$A=A'\cup B',$$

and

$$F(A')=B\pmod {D_{\omega_2}},\qquad F(B')=C\pmod {D_{\omega_2}}.$$

Consistency conjecture. The property $(*)$ is consistent with ZFC.

The source presents this as a consistency assertion connected with quantifier elimination for the Boolean-algebraic structure coding the monadic theory of $\omega_2$. The parser gives no resolution.