Partial-order conjecture for ribbon rational homology cobordism

A ribbon $\mathbb{Q}$-homology cobordism is a rational homology cobordism between closed, oriented, connected 3-manifolds admitting a handle decomposition with only 1- and 2-handles.

Ribbon rational homology cobordism conjecture. If there exists a ribbon $\mathbb{Q}$-homology cobordism from $Y_0$ to $Y_1$ and one from $Y_1$ to $Y_0$, then $Y_0$ and $Y_1$ are diffeomorphic; equivalently, ribbon $\mathbb{Q}$-homology cobordism is a partial order.

This is presented as the 3-manifold analogue of Gordon's ribbon-concordance conjecture. The general statement remains open.