  We establish a kind of "degree zero Freudenthal Gm-suspension theorem" in
motivic homotopy theory. From this we deduce results about the conservativity
of the P^1-stabilization functor.
  In order to establish these results, we show how to compute certain pullbacks
in the cohomology of a strictly homotopy invariant sheaf in terms of the
Rost--Schmid complex. This establishes the main conjecture of [BY18], which
easily implies the aforementioned results.
