On the intersection of free subgroups in free products of groups

Let (Gi | i I) be a family of groups, let F be a free group, and let $G = F \ast \mathop{\text{\Large $*$}}_{i\in I} G_i,$ the free product of F and all the Gi. Let $\mathcal{F} $ denote the set of all finitely generated subgroups H of G which have the property that, for each g G and each i I, $H \cap G_i^{g} = \{1\}.$ By the Kurosh Subgroup Theorem, every element of $\mathcal{F}$ is a free group. For each free group H, the reduced rank of H, denoted r(H), is defined as $\max \{\rank(H) -1, 0\} \in \naturals \cup \{\infty\} \subseteq [0,\infty].$ To avoid the vacuous case, we make the additional assumption that $\mathcal{F}$ contains a non-cyclic group, and we define $ \begin{eqnarray} \upp:= \sup \bigg\{\frac{\barr(H\cap K)} {\barr (H) \cdot\barr(K)} : H, K \in \mathcal{F} \text{ and} \barr (H) \cdot \barr(K) \neq 0\bigg\} \in [1,\infty]. \end{eqnarray}$ We are interested in precise bounds for $\upp$. In the special case where I is empty, Hanna Neumann proved that $\upp$ [1,2], and conjectured that $\upp$ = 1; fifty years later, this interval has not been reduced. With the understanding that /(-2) is 1, we define $ \fun:= \max\bigg\{\frac{\abss{L}}{ \abss{L}-2} : \text{$L$ is a subgroup of $G$ and } \abss{L} \ne 2 \bigg\} \in[1,3].$ Generalizing Hanna Neumann's theorem we prove that $\upp \in [\fun, 2\fun]$, and, moreover, $\upp = 2\fun$ whenever G has 2-torsion. Since $\upp$ is finite, $\mathcal{F}$ is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that $\upp = \fun$ whenever G does not have 2-torsion.