No zeros of $\zeta$ with $\tfrac12<\operatorname{Re} s<1$ above height $2$

Let $\zeta$ denote the Riemann zeta function. The assertion is that $\zeta$ has no zero strictly to the right of the critical line inside the critical strip, at height larger than $2$:

This is open: it is the Riemann hypothesis with the two parts that are unconditionally known removed. Indeed, non-vanishing for $\operatorname{Re} s\ge 1$ is classical (Mathlib's riemannZeta_ne_zero_of_one_le_re), and non-vanishing in the low-lying rectangle $0<\operatorname{Re} s<1$, $|\operatorname{Im} s|\le 2$ is an explicit estimate on the completed zeta function. Together with those two facts, the statement above is equivalent to the half-plane form of the Riemann hypothesis, $\operatorname{Re} s>\tfrac12\Rightarrow\zeta(s)\neq 0$, and hence to the Riemann hypothesis itself.

Phrasing the remaining open problem this way isolates exactly the region where the difficulty lies: a bounded-real-part, unbounded-height region on one side of the critical line. Any progress of zero-density or zero-free-region type applies to it directly.

Formalization note. riemannZeta is Mathlib's zeta function; |s.im| is the absolute value of the imaginary part of $s$.