# Is there a generalization of Araki-Lieb-Thirring inequality for four matrices?

It is known that $$Tr[(AB)^n] \leq Tr (A^n B^n)$$ for $A$, $B$ positive semi-definite matrices. I am looking of a generalization of it for a product of $4$ positive semi-definite matrices of the the form, $$Tr [(ABCD)^n] \leq Tr [(AB)^{2n} ] Tr [(DC)^{2n}]$$ Is there any such inequality? Can someone suggest an approach if the above thing can be proved or disproved?