The continuity conjecture for p-rationality levels

Let $p$ be a prime, $G$ a finite group, and let $\alpha\in\mathbb{Z}^{\geq 2}$. The $p$-rationality level of a character is the integer determined by the $p$-part of its conductor. Continuity conjecture. If $G$ has an irreducible $p'$-degree character of $p$-rationality level $\alpha$, then $G$ has irreducible $p'$-degree characters of every level from $2$ to $\alpha$.

The paper proves this conjecture for $p=2$, while the general case remains open in the supplied text.