Before revision:  As an example of the applicability of our construction, we obtain a new \tilde{\Omega}(1/\epsilon^4) sampling lower bound for testing the Schatten-1 norm with a \epsilon  ^{1.5} gap, extending a result of Balcan, Li, Woodruff, and Zhang [SODA'19].
After revision:  As an example of the applicability of our construction, we obtain a new \tilde{\Omega}(1/\epsilon^4) sampling lower bound for testing the Schatten-1 norm with a \epsilon n ^{1.5} gap, extending a result of Balcan, Li, Woodruff, and Zhang [SODA'19].