matrices - On a positivity of a matrix with trace entries.

Some basic observations lead me to ask the following quesiton

Let $A_1, cdots, A_m$ be $ntimes n$ complex matrices. For positive integer $kge 1$, show
$$left(begin{array}{cccc}Tr{(A_1^*A_1)^k}&Tr{(A_1^*A_2)^k}&cdots &Tr{(A_1^*A_m)^k}\Tr{(A_2^*A_1)^k}&Tr{(A_2^*A_2)^k}&cdots &Tr{(A_2^*A_m)^k}\cdots&cdots&cdots&cdots\Tr{(A_m^*A_1)^k}&Tr{(A_m^*A_2)^k}&cdots &Tr{(A_m^*A_m)^k}
end{array}right)$$
is positive semidefinite.

Remark

1). When $m=2$, it suffices to show $|Tr{(A_1^*A_2)^k}|^2le Tr{(A_1^*A_1)^k}cdot Tr{(A_2^*A_2)^k}$, which is a consequence of a unitarily invariant norm inequality appeared in p.81 of X.Zhan, Matrix inequalities, Springer, 2002.

2). It is easy to show $$left(begin{array}{cccc}(Tr{A_1^*A_1})^k&(Tr{A_1^*A_2})^k&cdots &(Tr{A_1^*A_m})^k\(Tr{A_2^*A_1})^k&(Tr{A_2^*A_2})^k&cdots &(Tr{A_2^*A_m})^k\cdots&cdots&cdots&cdots\(Tr{A_m^*A_1})^k&(Tr{A_m^*A_2})^k&cdots &(Tr{A_m^*A_m})^k
end{array}right)$$
is positive semidefinite, since it is $k$ Hadamard product of a Gram matrix.