The zero-Euler-characteristic bounding conjecture for Haken 3-manifolds

Let $M^{3}$ be a closed, oriented Haken $3$-manifold. A compact, oriented Haken $4$-manifold $W^{4}$ is required to have boundary $\partial W^{4}=M^{3}$, with boundary understood to be $\pi_1$-injective. Zero-Euler-characteristic bounding conjecture. There is such a $W^{4}$ satisfying

$$\chi(W^{4})=0.$$

This conjecture concerns bounding Haken $3$-manifolds by Haken $4$-manifolds with minimal specified Euler characteristic; the paper uses related building blocks and constructions but does not establish the assertion in full generality.