Fractional list and correspondence packing conjectures

Let $G$ be a graph. Write $\chi_\ell^\bullet(G)$ and $\chi_c^\bullet(G)$ for its fractional list and fractional correspondence packing numbers, and write $\chi_\ell(G)$ and $\chi_c(G)$ for the corresponding ordinary list and correspondence chromatic numbers.

Fractional packing conjectures. There exists a constant $C>0$ such that, for every graph $G$,

$$\chi_\ell^\bullet(G)\leq C\cdot\chi_\ell(G),$$

and there exists a constant $C>0$ such that, for every graph $G$,

$$\chi_c^\bullet(G)\leq C\cdot\chi_c(G).$$

The fractional parameters provide relaxations of the integral packing numbers, and the paper proves an upper bound in terms of maximum degree for the fractional versions. Whether each fractional parameter is bounded by a universal constant multiple of its integral counterpart remains open.