Wise's conjecture on locally indicable towers and vanishing $L^2$-Betti number

Let $Y$ be a finite connected $2$-dimensional $CW$-complex. A connected tower with target $Y$ is a finite sequence of covering maps and embeddings

$$X=Y_n\xrightarrow{i_n}\overline{Y_n}\xrightarrow{p_n}Y_{n-1}\xrightarrow{i_{n-1}}\overline{Y_{n-2}}\xrightarrow{p_{n-2}}\cdots\xrightarrow{p_2}Y_1\xrightarrow{i_1}\overline{Y_1}\xrightarrow{p_1}Y_0=Y.$$

Call such a tower non-positive when its source is contractible or has non-positive Euler characteristic. Wise's conjecture. The following statements are equivalent:

1. Every connected tower with $Y$ as target is non-positive.
2. Every connected tower with $Y$ as target is non-positive, provided that the fundamental group of each space appearing in it is locally indicable.
3. The group $\pi_1(Y)$ is locally indicable and

$$b^{(2)}_2(\widetilde Y)=b^{(2)}_2(\widetilde Y;\mathcal N(\pi_1(Y)))=0.$$

The conjecture is a modification of an earlier geometric characterization associated with Wise and Chemtov. It would connect non-positivity of towers, local indicability, and vanishing of the top $L^2$-Betti number; the supplied source does not resolve it.