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\end{equation}  Now that the uncertainty due to the Higgs boson mass dependence is no longer relevant, that the uncertainty due to the top-quark mass dependence is negligible, and the pQCD scale uncertainty from the latest {\rm N_3LO} calculations~has dropped to 0.0002, this method allows access to high precision on $\alpha_{\rm s}$. As shown in Eq.~\ref{eq:Rl}, $R_\ell$ was measured at LEP with a relative uncertainty is 0.12\%. This precision is expected to be improved to better than $10^{-5}$ at TLEP. The LEP experimental error of 0.0038 will scale accordingly.\\  {\em A reasonable target for the measurement of $\alpha_s (m^2_{\rm Z})$ with a run at the Z pole with TLEP is therefore a precision better than of  0.0002.}