Thom's finiteness conjecture for polynomial-map topological types

Let $cK$ be either $cC$ or $cR$, and let $cPKknp$ denote the space of degree-$d$ polynomial maps from $cK^n$ to $cK^p$, with topological type defined by composition with homeomorphisms on source and target. Thom's finiteness conjecture. For every positive integer $d$ and every positive integer $n$, the space $cPKkno$ gives rise to at most finitely many topological types. The conjecture is the central finiteness claim motivating the survey; the provided text does not establish whether it has been resolved.