Integral congruence conjecture for skew-symmetric matrices

Let $A$ and $H$ be skew-symmetric matrices with integer entries. Assume that $H$ is nondegenerate, $\mathop{\fam0 rk} A\leq\mathop{\fam0 rk} H$, and every $\mathop{\fam0 rk} H\times\mathop{\fam0 rk} H$ minor of $A$ is divisible by $\det H$.

Integral congruence conjecture. There is an integer matrix $Y$ such that

$$A=Y^T H Y.$$

The divisibility condition is automatic when $\mathop{\fam0 rk} A<\mathop{\fam0 rk} H$. The source presents this matrix assertion as the ingredient needed for the corresponding characterization of $\mathbb Z$-embeddings of $k$-complexes; its resolution status is not specified here.