Homotopy stability of spaces of non-resultant systems of bounded multiplicity with real coefficients

We continue our study of the topology of the spaces of $m$ tuples of real polynomials with common degree $d$ and without common roots of multiplicity $n$, and in particular their stability properties with respect to $d$. In an earlier paper we have proved a homotopy stability result and determined the stable homotopy types of such spaces in the case $mn>=4$. In the case $mn=3$ we could only prove stability in homology. In this paper we prove the corresponding homotopy result for the case $mn=3$.