Hanna Neumann conjecture

Statement
Let $$H$$ and $$K$$ be finitely generated nontrivial subgroups of a free group. Then:

$$rank(H \cap K) - 1 \le (rank(H) - 1 )(rank(K) - 1)$$

Neumann's own work
Hanna Neumann, in her paper On the intersection of finitely generated free groups, proved the following:

Let $$H$$ and $$K$$ be finitely generated nontrivial subgroups of a free group. Then:

$$rank(H \cap K) - 1 \le 2(rank(H) - 1 )(rank(K) - 1)$$

She then asked whether the factor of 2 can be dropped.

Offshoots
Myasnikov has raised the following question:

Let $$m$$, $$n$$ be positive integers, and $$H$$ and $$K$$ nontrivial finitely generated subgroups of a free group such that $$rank(H) = n$$ and $$rank(K) = m$$. Which numbers between 1 and $$(n-1)(m-1)$$ can be realized as $$rank(H \cap K) - 1$$? In particular, can $$(n-1)(m-1) - 1$$ be realized?