The upper-envelope switching-point conjecture for polynomials

Let $f_1,\ldots,f_n$ be polynomials of degree at most $d$, and define

$$g(x)=\max\{f_1(x),\ldots,f_n(x)\}.$$

Upper-envelope switching-point conjecture. The function $g$ has at most $d(n-1)$ points at which it switches between a pair of the polynomials $f_i$.

This bound would imply that the partition arising from the upper envelope has at most $d(n-1)+1$ segments, allowing the continuous splitting-necklace theorem to be reduced to the convex equipartition theorem. The statement is not marked as resolved in the supplied source.