Bochi–Fayad conjecture on resistant pairs of matrices

Let $\mathcal{H}$ denote the set of all $2 \times 2$ real matrices with unit determinant and unequal real eigenvalues, and let $\mathcal{R}$ denote the set of all $2 \times 2$ real matrices with unit determinant and non-real eigenvalues. A pair $(A_1,A_2)$ of $2 \times 2$ real matrices is resistant if there exist $c,\varepsilon,\gamma>0$ such that, for every $n\geq 1$ and every choice of $i_1,\ldots,i_n\in\{1,2\}$, at most $\varepsilon n$ of the integers $i_k$ being equal to $2$ implies

$$\|A_{i_1}\cdots A_{i_n}\|\geq ce^{\gamma n}.$$

Bochi–Fayad conjecture. The set of all resistant pairs $(H,R)\in\mathcal{H}\times\mathcal{R}$ has full Lebesgue measure. Partial results are known, and explicit resistant pairs can be constructed in special cases, but the full-measure assertion remains open.