It is straightforward to prove that the product $A_1 A_2\cdots A_k$ of $k$ (different) symmetric, real, positive semidefinite matrices is also symmetric if $A_i A_j=A_j A_i$ for all $i,j$. Moreover, it is well-known that for the case $k=2$, this pairwise commutativity condition is also a sufficient condition for symmetry of the matrix product.
My question is the following: Is there a result for $k>2$ concerning sufficient conditions for the symmetry of the product $A_1 A_2\cdots A_k$ of $k$ symmetric, real, positive semidefinite matrices? I have a set of $k$ matrices whose product I know is symmetric, but I would like to know if there's a result in the literature placing any restrictions on the individual matrices $A_i$. I suspect it has to do with pairwise commutativity, but have not been able to figure it out.
Thanks in advance for any insights!