The resist-impurities conjecture for pairs of $2\times2$ matrices

Let $\mathcal{H}\subset\mathit{SL}_2(\mathbb{R})$ be the set of matrices with distinct real eigenvalues, and let $\mathcal{E}\subset\mathit{SL}_2(\mathbb{R})$ be the set of matrices with distinct non-real eigenvalues. A pair $(H,R)\in\mathcal{H}\times\mathcal{E}$ resists impurities if there exist constants $\varepsilon,\lambda>0$ such that every product $A_n\cdots A_1$ of $H$ and $R$ containing at most $\varepsilon n$ instances of $R$ satisfies

$$\|A_n\cdots A_1\|\geq e^{\lambda n}.$$

The resist-impurities conjecture. The set of pairs $(H,R)\in\mathcal{H}\times\mathcal{E}$ that resist impurities has full Lebesgue measure in $\mathcal{H}\times\mathcal{E}$. This conjecture was introduced in earlier work and has partial results, but the stated full-measure claim remains open.