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.