matrices - What's known about the 3rd coefficient in the BMV conjecture?

The most appealing statement of the Bessis-Moussa-Villani conjecture is as follows:

Conjecture: For all Hermitian positive semidefinite $ntimes n$ matrices $A$ and $B$,
and all positive integer $m$, the polynomial function
$$t in mathbb{R}mapsto g(t) equiv tr[(A + t B)^m] =
sumlimits_{
k=0}^m
a_kt^k$$
has only nonnegative coefficients $a_k, k=1,cdots,m$.

Most recent and past research concerns on the quantities $m$ and $n$. What about trying to prove the conjecture in the following way: first show that $a_mge0$, $a_0ge0$, $a_{m-1}ge0$, $a_1ge0$, $a_{m-2}ge0$, $a_2ge0$. And then go on to show $a_3ge 0$ (and so is $a_{m-3}ge 0$)...

But it seems difficult to show $a_3ge 0$. Can anyone share some idea on this particular coefficient?

UPDATED
What is the largest term in $S_{2m,m}(AB)$ ? More precisely, consider the word in two positive definite letters $A^{k_1}BA^{k_2}Bcdots A^{k_m}B$ , where $(k1,k2,cdots,k_m)$ is a pair of nonnegative integer solution of $k_1+k_2+cdots+k_m=m$ . Is it true that $tr(A^{k_1}BA^{k_2}Bcdots A^{k_m}B)le tr(A^mB^m)$ ?