2 problems

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…

Huynh–Joos–Wollan conjecture. Even for $k>2$, the family of $(\Gamma_1,\Gamma_2,\ldots,\Gamma_k)$-non-zero cycles has the half-integral Erdős–Pósa property.