Lossless Strichartz estimates on rectangular tori over short time intervals

We prove lossless Strichartz estimates at the critical exponent $q_c=\frac{2(n+1)}{n-1}$ and the endpoint exponent pair $(2,\frac{2(n-1)}{n-3})$ for the Schrödinger equation on rectangular tori of dimension $n-1$ with frequency localized initial data on small time windows with length depending on the frequency parameter $\lambda \gg 1$.