  In \cite{PRA} and \cite{SSM} the orthonormal Strichartz estimates for the
Schr\"odinger equation associated with the Dunkl Laplacian and the
Dunkl-Hermite operator are obtained. In this article, we prove a necessary
condition on the Schatten exponent for the above orthonormal Strichartz
estimates, which turns out to be optimal for the Schr\"odinger equations
associated with Laplacian and Hermite operator as a special case.
