Defect-normalizer conjecture for the p-rationality level of height-zero characters

Let $p$ be a prime, $G$ a finite group, and let $B$ be a $p$-block of $G$ with defect group $D$. Let $u$ be the $p$-adic valuation, let $c(\rho)$ denote the conductor of the values of a character $\rho$, and define its $p$-rationality level by

$$\mathrm{lev}(\rho):=\nu(c(\rho)).$$

For a character of $G$, write $\chi_{\mathbf{N}_G(D)}$ for its restriction to the defect normalizer. Defect-normalizer conjecture. If $\chi$ is a height-zero character in $B$ with $\mathrm{lev}(\chi)\geq 2$, then

$$\mathrm{lev}(\chi)=\mathrm{lev}(\chi_{\mathbf{N}_G(D)}).$$

This proposes that the $p$-rationality level of a height-zero character is detected inside the normalizer of a defect group, providing a global-local principle for character rationality. Its status is not resolved in the supplied text.