The multiple-constraint Erdős–Pósa conjecture

Let $t\geq 1$ be a positive integer, and let

$$\Gamma=\bigoplus_{i=1}^t\Gamma_i,$$

where each $\Gamma_i$ is a group. A $\bigoplus_{i=1}^t\Gamma_i$-labeled graph is a graph whose edges are labeled by this direct sum, and a cycle is $(\Gamma_1,\dots,\Gamma_t)$-non-zero when it is non-zero in every component.

Multiple-constraint Erdős–Pósa conjecture. The set of all $\bigoplus_{i=1}^t\Gamma_i$-labeled graphs has the half-integral Erdős–Pósa property for $(\Gamma_1,\dots,\Gamma_t)$-non-zero cycles. Moreover, the Erdős–Pósa function does not depend on the choice of $\Gamma_1,\dots,\Gamma_t$.

This would imply the half-integral Erdős–Pósa property for cycles satisfying more than two constraints. The conjecture proposes a uniform Erdős–Pósa function across all choices of the component groups; its resolution is not given in the source.