The two-generator conjecture for the mod 2 cohomology of $BE_8$

Let $E_8$ be the exceptional Lie group, and let $BE_8$ be its classifying space. Write

$$\rho_8=\operatorname{Ad}_{E_8}\colon E_8\longrightarrow SO(248)$$

for the adjoint representation, and let $w_{128}(\rho_8)$ denote its 128-th Stiefel–Whitney class. The two-generator conjecture for $BE_8$. The mod 2 cohomology $H^*(BE_8;\mathbb Z/2)$ has two generators as an algebra over the Steenrod algebra; the first has degree 4, and the second is $w_{128}(\rho_8)$. The structure of the mod 2 cohomology of $BE_8$ is largely unknown, and this conjecture predicts the number and specific characteristic-class representative of its Steenrod-algebra generators.