Budan's theorem gives an upper bound for the number of real roots of a real polynomial in a given interval $(a,b)$. This bound is not sharp (see the example in Wikipedia).

My question is the following: let us suppose that Budan's theorem tells us "there are $0$ or $2$ roots in the interval $(a,b)$" (or more generally "there are $0$, $2$, ... $2n$ roots"). Let us suppose also that there are actually no roots in the interval. Is it true that there are $2$ roots whose real part lies in the interval $(a,b)$?

Thank you!