Theorem 3.1.6. of the paper [Critical points of real polynomials ...][1] 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 <em>above</em>, as all the previous cases handle degrees $d\ge5$. 


  [1]: https://www.researchgate.net/profile/Eugenii-Shustin-2/publication/267655438_Critical_points_of_real_polynomials_subdivisions_of_Newton_polyhedra_and_topology_of_real_algebraic_hypersurfaces/links/593ec9480f7e9bf167c5c86c/Critical-points-of-real-polynomials-subdivisions-of-Newton-polyhedra-and-topology-of-real-algebraic-hypersurfaces.pdf?origin=publicationDetail&_sg%5B0%5D=q9Oh1_m3jOK6GbcVqAfUCbUZQcUhzBdCSuiy3C4-M4tptZBRj9TUFvwMlxungIYZGBJCN448_Ym-sIGtnLa7Tg.4tpQfI3YawFUZmb-f-azz2eQ4vhkYXKkq72yY3dxP10EjQoQVfzrjiDaqocMD5tauPGPA_EHk65Awgyzqi5H6A&_sg%5B1%5D=SGk8BZqoH6nDDCqLNnHvxuM-t9dfzYMqDyjU0p1TCYgtjQarC-uAAejb8-TZRRJc3tDQhpv1GlfrHxvmiaWwho-UDx5azx9mjQ7Oto0W3pHr.4tpQfI3YawFUZmb-f-azz2eQ4vhkYXKkq72yY3dxP10EjQoQVfzrjiDaqocMD5tauPGPA_EHk65Awgyzqi5H6A&_iepl=&_rtd=eyJjb250ZW50SW50ZW50IjoibWFpbkl0ZW0ifQ%3D%3D