Wise's conjecture on non-positive immersions for contractible complexes

Let $X$ be a contractible complex. Wise's conjecture. The complex $X$ has (weak) non-positive immersions. Here, non-positive immersions means that for every immersion $Y\looparrowright X$ of a compact, connected 2-complex $Y$, either $\chi(Y)\le 0$ or $\pi_1 Y=1$; weak non-positive immersions means that $\chi(Y)\le 1$ for every compact, connected $Y\looparrowright X$. The paper's abstract states that it provides examples of contractible complexes failing to have non-positive immersions and weak non-positive immersions, answering Wise's conjecture in the negative.