Say that a monic univariate polynomial in an indeterminate $x$ with real coefficients of degree $d$ has positive coefficients if each of its coefficients of $x^d,\ldots,x^1,x^0$ are (strictly) positive.

For $f$ a univariate polynomial with real coefficients, if there exists a positive integer $m_0$ such that its $m_0$th power $f^{m_0}$ has positive coefficients, then does there exist a positive integer $m_1$ such that for each positive integer $m$ at least $m_1$, the $m$th power $f^m$ has positive coefficients?