Infinite-group Erdős–Pósa characterization for allowable A-paths

Let $m$ and $\tau$ be positive integers and, for each $i\in[m]$, let $\Gamma_i$ be an abelian group and let $\Omega_i$ be a subset of $\Gamma_i$ with $|\Omega_i|\leq \tau$. Set

$$\Lambda=(\Gamma_1-\Omega_1)\times\dots\times(\Gamma_m-\Omega_m),$$

and let $\mathcal{F}$ be the family of $\Lambda$-allowable $A$-paths.

Infinite-group characterization conjecture. The family $\mathcal{F}$ satisfies the half-integral Erdős–Pósa property, and it satisfies the Erdős–Pósa property if and only if $\Lambda$ satisfies the Erdős–Pósa condition.

The conjecture extends the finite-abelian-group characterization proved in the paper to products of abelian groups that may be infinite, and it addresses the broader setting posed by Gollin et al. The analogous characterization for cycles is known, while the proposed extension for allowable $A$-paths remains open because the proof tools used in the finite case depend on finiteness.