  We report a sensitive study of $D_s^+ \to \tau^+ \nu_{\tau}$ decay via
$\tau^+ \to e^+ \nu_e \bar{\nu}_{\tau}$ with an integrated luminosity of 1
ab$^{-1}$ at the center-of-mass energy of $4.009$ GeV at a future Super Tau
Charm Facility (STCF). Under the help of the fast simulation software package,
the statistical sensitivity for the absolute branching fraction of $D_s^+ \to
\tau^+ \nu_{\tau}$ is determined to be $2\times10^{-4}$. Combining with our
previous prospect of $D_s^+ \to \mu^+ \nu_{\mu}$, the ratio of the branching
fractions for $D_s^+ \to \tau^+ \nu_{\tau}$ over $D_s^+ \to \mu^+ \nu_{\mu}$
can achieve a relative statistical precision of 0.5%, which will provide the
most stringent test of the $\tau$-$\mu$ lepton flavor universality in heavy
quark decays. Taking the decay constant $f_{D_s^+}$ from lattice QCD
calculations or the CKM matrix element $|V_{cs}|$ from the CKMfitter group as
an input, the relative statistical uncertainties for $|V_{cs}|$ and $f_{D_s^+}$
are estimated to be 0.3% and 0.2%, respectively.
