Daemi–Lidman–Vela-Vick–Wong rational homology ribbon cobordism partial-order conjecture

Let $W_0$ and $W_1$ be closed, connected, oriented $3$-manifolds. A rational homology ribbon cobordism from $W_0$ to $W_1$ is a ribbon cobordism $W$ from $W_0$ to $W_1$ such that the inclusions $W_0\hookrightarrow W$ and $W_1\hookrightarrow W$ induce isomorphisms on rational homology groups. Write $W_0\leq W_1$ when such a cobordism exists. Daemi–Lidman–Vela-Vick–Wong's conjecture. The preorder on the set of homeomorphism classes of closed, connected, oriented $3$-manifolds given by rational homology ribbon cobordism is a partial order: if

$$W_0\leq W_1\quad\text{and}\quad W_1\leq W_0,$$

then $W_0$ and $W_1$ are homeomorphic. The paper proves this conjecture for all closed, connected, oriented $3$-manifolds, extending earlier results for aspherical manifolds and the irreducible case.