On the Grassmanian image of submanifolds $F^n\subset E^{n+m}$ in which codimension does not exceed the dimension

A. A. Borisenko's hypothesis is studied: every tangent space of the Grassman image of a regular submanifold $F^n\subset E^{n+m}$ contains a two-dimensional plane $\pi$ such that the sectional curvature of the Grassman manifold $G_{n,n+m}$ in $\pi$ is less or equal to $1$.