The existence of an aspherical 4-manifold with Euler characteristic one

Let $M$ be an oriented closed aspherical $4$-manifold, and let $\chi(M)$ denote its Euler characteristic.

Aspherical 4-manifold Euler-characteristic problem. There exists an oriented closed aspherical $4$-manifold $M$ with

$$\chi(M)=1.$$

This is presented as an open problem in $4$-dimensional topology and is used to motivate the preceding question about fillings of aspherical $3$-manifolds. No example or obstruction is supplied in the source.