Finite-type conjecture for symmetric powers of Rep(C_2)-complexes

Let $G=C_2$. A $\operatorname{Rep}(C_2)$-complex is a $C_2$-space built by attaching cells of the form $D(\mathbf R^{p,q})$, where $p\geq q\geq0$; it is of finite type if, for each $n\geq0$, it has finitely many cells with $p-q=n$. Finite-type conjecture. The symmetric powers of a $\operatorname{Rep}(C_2)$-complex are $\operatorname{Rep}(C_2)$-complexes. Moreover, if the original complex is of finite type, then all of its symmetric products are of finite type. In particular, the equivariant Eilenberg--MacLane spaces $K(\underline{\mathbf Z},V)$ are $\operatorname{Rep}(C_2)$-complexes of finite type. The conjecture concerns the cellular structure of symmetric powers and, in particular, the finite-type structure of these equivariant Eilenberg--MacLane spaces; the preceding context relates it to freeness results for Bredon cohomology of finite-type $\operatorname{Rep}(C_2)$-complexes.