Classification of primitive depth-two zeta-like tuples for q=2

Let $q=2$, and consider primitive tuples of depth $2$. Depth-two classification conjecture for q=2. The zeta-like, equivalently eulerian, primitive tuples are exactly

$$(1,1),\quad(1,3),\quad(3,5),\quad(2^n-1,2^n),\quad(2^n,2^{n+1}+2^n-1).$$

The source says its theorems prove that all listed tuples are eulerian, while the converse remains conjectural; numerical data supported the claim up to weight $128$.