[Plevnik (2016)](http://www.ams.org/mathscinet-getitem?mr=3552962) [(pdf)](http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/Vol47_2016_3_ART09.pdf) found the counterexample
$$
A = \begin{bmatrix}76&0&0\\0&0&0\\0&0&1\end{bmatrix}
\qquad\text{and}\qquad
B = \begin{bmatrix}20&-14&13\\-14&2880&3100\\13&3100&3380\end{bmatrix}
$$
with
$$
\operatorname{tr} A^4BAB^4 = 7608677695167720100 > 7566365725138281700 = \operatorname{tr} A^5B^5.
$$
(To get one with $A$ positive definite, replace entry $A_{22}$ by $0.01$ and obtain, unless I miscalculated,
$$
\operatorname{tr} A^4BAB^4 = 7.58468087608\times10^{18} > 7.56636572557\times10^{18} = \operatorname{tr} A^5B^5.)
$$