Theorem 3.1.6. of the paper Critical points of real polynomials ... by Shustin seems to claim that $5$ minima are possible. However, the "proof" given in Section 3.13 merely says "Assume that $d \le 4$. Then all the index distributions except [...] can be easily realized as it was explained above".
I don't know what the author means by above, as all the previous cases handle degrees $d\ge5$.