Canonical embedding isomorphism conjecture for spherical latin bitrades

Let $(W,B)$ be a spherical latin bitrade, and let $\mathcal{A}_W^*$ and $\mathcal{A}_B^*$ denote the canonical abelian groups into which the white and black trades embed, respectively; write $\mathcal{A}_W$ and $\mathcal{A}_B$ for the corresponding groups without the distinguished embedding data. Canonical embedding isomorphism conjecture. If $(W,B)$ is a spherical latin bitrade, then

$$\mathcal{A}_W^*\cong\mathcal{A}_B^* \quad\text{and}\quad \mathcal{A}_W\cong\mathcal{A}_B.$$

Although the minimal embeddings of the two trades can differ in general, the conjecture asserts that their canonical embeddings, and the associated abelian groups, coincide up to isomorphism for spherical latin bitrades.