Projected embeddings

An obvious necessary condition for a map $f: N\to M$ to lift to an embedding $N\to M\times\mathbb R^k$ is the existence of a $\mathbb Z_2$-equivariant map $\Delta_f\to S^{k-1}$, where $\Delta_f$ is the set of pairs $(x,y)\in N\times N$ such that $f(x)=f(y)$ and $x\ne y$. This condition is obviously not sufficient for the degree 3 covering $f: S^1\to S^1$ (with $k=1$), but M. Skopenkov proved that it is sufficient for maps $f$ of a trivalent graph into $\mathbb R^1$ (with $k=1$). Also, Haefliger proved in 1963 that it is sufficient in the case where $M$ is a point, $N$ is a smooth manifold and $2k\ge 3(\dim N+1)$.
We prove that the condition is sufficient when $f$ is a generic PL map or a generic smooth map, $n\le m$, $2(m+k)\ge 3(n+1)$ and $4n-3m\le k$, where $n=\dim N$, $m=\dim M$. In both cases the constructed lift will be only a piecewise-smooth embedding. When $f$ is a generic PL map, we can find a lift that is a PL embedding by some additional work. But if $f$ is a generic smooth map and we want the lift to be a smooth embedding, we must assume additionally that either $3n-2m\le k$ or $f$ has no singularities of type $\Sigma^{1,1}$.
One could try to prove these (or similar) results by some version of Haefliger's generalization of the Whitney trick. But, unfortunately, it does not work. We use a new kind of "Whitney trick", which in contrast to Haefliger's is described by an explicit formula.