Rigidity for tame cohomology over henselian pairs

Let $S$ be a henselian discrete valuation ring with maximal ideal $\mathfrak m_S=(\pi)$. Let $A$ be a separated essentially of finite type $S$-algebra such that $(A,I=(\pi)A)$ is a henselian pair. Let $i:\operatorname{Spec}(A/I)\to\operatorname{Spec}(A)$ be the corresponding inclusion, and let $F\in \operatorname{Sh}((\operatorname{Spec}(A)/S)_t)$. Rigidity conjecture. The restriction map

$$H^s(\operatorname{Spec}(A),F)\cong H^s(\operatorname{Spec}(A/I),i^*F)$$

is an isomorphism for all $s$.

This predicts invariance of tame cohomology under passage from the henselian pair to its closed fibre. The source gives no proof or resolution for this assertion.