On orthogonal projections of Nöbeling spaces

Suppose that $0\le k<\infty$. We prove that there is a dense open subset of the Grassmann space $\operatorname{Gr}(2k+1,m)$ such that the orthogonal projection of the standard Nöbeling space $N^m_k$ (which lies in $\mathbb R^m$ for sufficiently large $m$) to every $(2k+1)$-dimensional plane in this subset is $k$-soft and possesses the strong $k$-universal property with respect to Polish spaces. Every such orthogonal projection is a natural counterpart of the standard Nöbeling space for the category of maps.