Conjecture on the Hausdorff dimension of lower-spectral-radius discontinuities

Let $M_2(\mathbb{R})$ be the space of all $2\times2$ real matrices, and let

$$\underline{\varrho}\colon M_2(\mathbb{R})^2\to\mathbb{R}$$

be the lower spectral radius as a function on pairs of matrices. Discontinuity-dimension conjecture. The set of discontinuities of $\underline{\varrho}$ has Hausdorff dimension $8$. The ambient space $M_2(\mathbb{R})^2$ is eight-dimensional; the preceding theorem gives related generic discontinuity results on an open subset, but the stated dimension assertion remains open.