Positive-measure discontinuity conjecture for the lower spectral radius

Consider the lower spectral radius function

$$\check{\varrho}\colon\mathit{GL}_2(\mathbb{R})^2\to\mathbb{R}_+.$$

Positive-measure discontinuity conjecture. The set of discontinuities of $\check{\varrho}$ has positive Lebesgue measure. This is presented as a weaker version of the resist-impurities conjecture and remains open.