  A rigidity property for the homotopy invariant stable linear framed
presheaves is established. As a consequence a variant of Gabber rigidity
theorem is obtained for a cohomology theory representable in the motivic stable
homotopy category by a $\phi$-torsion spectrum with $\phi\in\mathrm{GW}(k)$ of
rank coprime to the (exponential) characteristic of the base field $k$. It is
shown that the values of such cohomology theories at an essentially smooth
Henselian ring and its residue field coincide. The result is applicable to
cohomology theories representable by $n$-torsion spectra as well as to the ones
representable by $\eta$-periodic spectra and spectra related to Witt groups.
