Bochnak–Kucharz approximation conjecture for odd-dimensional real algebraic sets

Let $n$ be a positive odd integer, and let $X$ be a nonsingular compact connected real algebraic set of dimension $n$. A regular map means a map given by regular functions on $X$. Bochnak–Kucharz approximation conjecture. Every continuous map from $X$ to $\mathbb{S}^n$ is homotopic to a regular one.

The conjecture concerns approximation of continuous maps by algebraic maps. It is known to hold for $n=1$, but the paper states that it is disproved for all $n\equiv 1\pmod 4$ with $n>1$; its status for the remaining odd dimensions is not settled here.