Realizable ranks of joins and intersections of subgroups in free groups

The famous Hanna Neumann Conjecture (now the Friedman-Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups $H$ and $K$ of a non-abelian free group. It is an interesting question to `quantify' this bound with respect to the rank of $H\vee K$, the subgroup generated by $H$ and $K$. We describe a set of realizable values $(rk(H\vee K),rk(H\cap K))$ for arbitrary $H$, $K$, and conjecture that this locus is complete. We study the combinatorial structure of the topological pushout of the core graphs for $H$ and $K$, with the help of graphs introduced by Dicks in the context of his Amalgamated Graph Conjecture. This allows us to show that certain conditions on ranks of $H\vee K$, $H\cap K$ are not realizable, thus resolving the remaining open case $m=4$ of Guzman's "Group-Theoretic Conjecture" in the affirmative. This in turn implies the validity of the corresponding "Geometric Conjecture" on hyperbolic $3$-manifolds with a $6$-free fundamental group. Finally, we prove the main conjecture describing the locus of realizable values for the case when $rk(H)=2$.