The difficulties with generalizations of the Golden-Thompson inequality to three matrices arise because the trace of a product of three positive symmetric matrices is in general not real; unlike the trace of the product of two positive symmetric matrices, which is real (and positive): ${\rm tr}\,e^B e^C={\rm tr}\,XX^t$ with $X=e^{B/2}e^{C/2}$ and $X^t$ the transpose of $X$.

So while ${\rm tr}\,Ae^{B+C}$ is a real number, ${\rm tr}\,Ae^Be^C$ is in general a complex number and no inequality between these two numbers is to be expected.

There do exist three-matrix generalizations of the Golden-Thompson inequality, but they take an entirely different form, see A survey of certain trace inequalities, equation 21.

The difficulties with generalizations of the Golden-Thompson inequality to three matrices arise because the trace of a product of three positive symmetric matrices is in general not positive; unlike the trace of the product of two positive symmetric matrices, which is positive: ${\rm tr}\,e^B e^C={\rm tr}\,XX^t=\sum_{n,m}X_{nm}^2>0$, with $X=e^{B/2}e^{C/2}$ and $X^t$ the transpose of $X$.

So while ${\rm tr}\,Ae^{B+C}$ is a positive number, ${\rm tr}\,Ae^Be^C$ can be negative.

There do exist three-matrix generalizations of the Golden-Thompson inequality, but they take an entirely different form, see A survey of certain trace inequalities, equation 21.