The half-integral reduction conjecture for labelled cycles

For an abelian group $\Gamma$, a subset $A\subseteq\Gamma$, and a $\Gamma$-labelled graph $(G,\gamma)$, let $\mathcal O^A_{G,\gamma}$ denote the set of cycles whose $\gamma$-values lie in $A$. Half-integral reduction conjecture. Let $s\geq 4$ be an integer. If there is a function $f\colon\mathbb{N}\to\mathbb{N}$ such that, for every $\Gamma$-labelled graph $(G,\gamma)$ and every positive integer $k$, there are either $k$ cycles in $\mathcal O^A_{G,\gamma}$ such that no vertex belongs to $s$ of them, or a hitting set for $\mathcal O^A_{G,\gamma}$ of size at most $f(k)$, then there is a function $f'\colon\mathbb{N}\to\mathbb{N}$ such that, for every $\Gamma$-labelled graph $(G',\gamma')$ and every positive integer $k$, there are either $k$ cycles in $\mathcal O^A_{G',\gamma'}$ such that no vertex belongs to three of them, or a hitting set for $\mathcal O^A_{G',\gamma'}$ of size at most $f'(k)$. This proposes that an Erdős–Pósa property with bounded vertex overlap $s\geq4$ implies its half-integral analogue with overlap three for cycles whose group values lie in $A$; the supplied text gives no resolution.