  As the first step in the direction of the Hopf conjecture on the
non-existence of metrics with positive sectional curvature on $S^2 \times S^2$
D.Gromoll and K.Tapp in [GT] suggested the following (Weak Hopf) conjecture (on
the rigidity of non-negatively curved metrics on $S^2 \times R^3$): "The
boundary $S^2\times S^2$ of the $S^2 \times B^3\subset S^2 \times R^3$ with an
arbitrary complete metric of non-negative sectional curvature contains a point
where a curvature of $S^2 \times S^2$ vanish". In this note we verify this.

Differential Geometry