The integral restriction conjecture for characters of finite groups

Let $G$ be a finite group, let $p$ be a prime, let $P$ be a Sylow $p$-subgroup of $G$, and let $\chi\in\operatorname{Irr}(G)$. For a subgroup $Q\le P$, write $\chi(1)_p$ for the $p$-part of $\chi(1)$. Integral restriction conjecture. The restriction $\chi_P$ is an integral linear combination of characters induced from subgroups $Q\le P$ such that

$$\chi(1)_p=|P:Q|.$$

The theorem preceding this conjecture proves the analogous assertion for finite $p$-solvable groups with a sum of induced characters. The conjecture asks whether the weaker integral-linear-combination statement holds for arbitrary finite groups; the authors note that the stronger statement is false for $\operatorname{PSU}(5,2)$ when $p=2$.