Wise's conjecture on towers and the second $L^2$-Betti number

Let $Y$ be a $2$-complex. Write $b^{(2)}_2(\widetilde Y;\mathcal N(\pi_1(Y)))$ for the second $L^2$-Betti number of its universal covering, with coefficients in the von Neumann algebra of $\pi_1(Y)$, and say that $Y$ has non-positive towers when every tower map from a finite, connected $2$-complex $X$ to $Y$ has either $X$ contractible or $\chi(X)\leq 0$. Wise's conjecture. The following are equivalent:

1. $b^{(2)}_2(\widetilde Y;\mathcal N(\pi_1(Y)))=0$.
2. $Y$ has non-positive towers.

This conjecture proposes a geometric characterization of the vanishing of the top $L^2$-Betti number of a $2$-complex. The paper studies this relationship and related characteristic-$p$ phenomena; the conjecture remains unresolved in the supplied source.