The conjecture on the 19th and 20th lattice invariants

Let $v$ be the invariant assigned to the lattices indexed by $1,\ldots,24$, with the indices $19$ and $20$ referring to the corresponding entries in the preceding table. Conjecture on the lattice invariants.

$$v(19)=5\quad\text{and}\quad v(20)=5.$$

The preceding bounds and product calculations establish several related values, while this assertion is used to derive $v(15)=4$ and the dimensions of the spaces ${\cal D}_3$, ${\cal D}_4$, and ${\cal D}_5$. The paper later notes that generalized Duke–Imamoğlu–Ikeda-lift calculations suggest the conjecture, but does not prove it.