Is there a trace inequality for the product of a sequence of hermitian postive definite matrices? Let $A$ and $B$ be two Hermitian matrices with positive eigenvalues. 
Let $k>0$ be a integer.
Let $P=(P_1,P_2,\dots,P_{2k})$ be a sequence of $k$ $A$s and $k$ $B$s in any given order.
Do we have
${\rm tr}\,\prod^{2k}_{i=1} P_i \leq {\rm tr}\,A^k B^k$ ?
 A: Although the general inequality is false (counterexamples can be found easily), for words where $A$ and $B$ occur in pairs, a stronger inequality is known. Indeed, a classic theorem of Ando and Hiai (1994) yields that for $A, B \ge 0$ and arbitrary reals $p_1,\ldots,p_k \ge 0$ we have
\begin{equation*}
  \text{tr}|A^{p_1}B^{p_1}\cdots A^{p_k}B^{p_k}| \le \text{tr}|A^{p_1+\cdots p_k}B^{p_1+\cdots+p_k}|,
\end{equation*}
where $|X|=(X^*X)^{1/2}$ denotes the matrix absolute value.

Remark. 
An easy way to obtain counterexamples for $3\times 3$ matrices similar to the one mentioned in Francois's answer is to use matrices of the form (the values are somewhat arbitrary, other choices also work):
\begin{equation*}
 X = \begin{bmatrix}a & b& 1\\ 0 & c & -2\\ 0 &0 &0\end{bmatrix},\quad
 Y = \begin{bmatrix}p & q& 1\\ 0 & r & -1\\ 0 &0 &-1\end{bmatrix},
\end{equation*}
then to define $A=XX^T$ and $B=YY^T$. Then, try to numerically minimize the quantity $g(a,b,c,p,q,r) := \text{tr}(A^5B^5)-\text{tr}(A^4BAB^4)$. Doing so, for instance in Mathematica, easily yields $g<0$, and thus, a desired counterexample. By playing around with different $X$ and $Y$, one can easily generate additional counterexamples.
A: A counterexample was apparently first found by Plevnik (2016) (pdf):
$$
A = \begin{bmatrix}76&0&0\\0&0&0\\0&0&1\end{bmatrix}
\qquad\text{and}\qquad
B = \begin{bmatrix}20&-14&13\\-14&2880&3100\\13&3100&3380\end{bmatrix}
$$
give
$$
\operatorname{tr} A^4BAB^4 = 7608677695167720100 > 7566365725138281700 = \operatorname{tr} A^5B^5.
$$
To get one with $A$ positive definite as requested, replace entry $A_{22}$ by $0.01$: the two traces become
$$
7584680876077508226.18611992\ \ > \ \ 7566365725573314229.03610008.
$$
