Conjecture on burying pairs of binary lattices in rank 3

Let $\ell_1$ and $\ell_2$ be binary $\mathbb{Z}$-lattices with equal discriminants

$$d\ell_1=d\ell_2.$$

Assume that the pair $(\ell_1,\ell_2)$ is buried in a genus of rank $3$. Burying conjecture. The pair $(\ell_1,\ell_2)$ is buried in rank $3$. The conjecture has been checked for $d\ell_1=d\ell_2\leq 3000$, but its general validity remains open.