The completeness conjecture for realizable join and intersection ranks

Let $F$ be a free group, and let $h,k,v,c$ be integers satisfying

$$2\le h\le k,\qquad 2\le v\le h+k,\qquad 0\le c\le (h-1)(k-1)+1.$$

For subgroups $H,K\le F$, the quantities $\operatorname{rk}(H)$, $\operatorname{rk}(K)$, $\operatorname{rk}(H\vee K)$, and $\operatorname{rk}(H\cap K)$ denote their ranks, with $H\vee K$ the subgroup generated by $H$ and $K$. Let $(a_i)$ be the sequence defined in Theorem~. The completeness conjecture. There exist subgroups $H,K\le F$ such that

$$\operatorname{rk}(H)=h,\qquad \operatorname{rk}(K)=k,\qquad \operatorname{rk}(H\vee K)=v,\qquad \operatorname{rk}(H\cap K)=c$$

if and only if $c\le a_i$ for $i=h+k-v$. The conjecture asserts that the realizable values described by Theorem~ are complete. The source notes that this conjecture subsumes Ivanov's open question about the extremal Hanna Neumann case, so it remains open.