Let $g: S^n \to R^n$ be a continuous odd function (i.e. $g(-x)=-g(x)$ for all $x$). The Borsuk-Ulam theorem implies that $g$ has a zero, i.e. there is an $x$ such that $g(x)=(0,0,...,0)$.

Suppose $g$ is (1,1,...,1) on the positive orthant (i.e. when all its $n$ arguments are non-negative) and (-1,-1,...,-1) on the negative orthant. Is this true that for every constant $r\in [-1,1]$, there is an $x$ such that $g(x)=(r,r,...,r)$?

For $n=1$, this is obviously true by the intermediate value theorem. Under what conditions is it true for $n>1$?

The question seems closely related to the Poincare-Miranda theorem, which is a generalization of the IVT to multi-dimensional cubes, but so far I haven't found the connection.