Kim–Livingston conjecture for rational homology cobordism

Let $\Theta^{\mathrm{top}}_\mathbb{Q}$ be the topological rational homology cobordism group of rational homology $3$-spheres, and let $W(\mathbb{Q}/\mathbb{Z})$ be the Witt group of nonsingular symmetric bilinear linking forms $L\colon A\times A\to\mathbb{Q}/\mathbb{Z}$ on finite abelian groups. The linking form defines a surjective homomorphism

$$\Theta^{\mathrm{top}}_\mathbb{Q}\to W(\mathbb{Q}/\mathbb{Z}).$$

Kim–Livingston conjecture. This homomorphism is an isomorphism.

The conjecture identifies topological rational homology cobordism classes of rational homology $3$-spheres with their linking-form Witt classes. The source describes it as a well-known conjecture and notes that surjectivity is known, while injectivity remains the unresolved part.