An affine property of Boolean functions on subspaces and their shifts

Let a Boolean function in $n$ variables be affine on an affine subspace of dimension $\lceil n/2 \rceil$ if and only if $f$ is affine on any its shift. It is proved that algebraic degree of $f$ can be more than $2$ only if there is no affine subspace of dimension $\lceil n/2 \rceil$ that $f$ is affine on it.