A zero-location inequality for Dirichlet $L$-functions

Let $\varrho:=\beta+i\gamma$ range over the zeros under consideration of Dirichlet $L$-functions. Zero-location conjecture. All zeros $\varrho:=\beta+i\gamma$ satisfy

$$\beta^2-\gamma^2<\frac14.$$

The source introduces this condition alongside Bentz's zero-free and nonvanishing assumption in the study of quadratic residues and prime distribution. No evidence of resolution is supplied.