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I discuss the possibility of estimating the shift $\delta\Delta\alpha(M_Z^2)$on the running of the EM-coupling to the $M_Z$ scale, induced by thediscrepancy $\Delta a_{\mu}$ between two precise determinations of the hadronicvacuum polarization contribution to $g_{\mu}-2$. It is shown that the size of$\Delta a_{\mu}$ implies rigorous bounds on the$\delta\Delta\alpha(M_Z^2)$-shift. Any extra contribution to this minimal shiftnecessarily depends on the specific shape of the underlying spectral function$\frac{1}{\pi}\Imm\Delta(t)$ responsible for the $\Delta a_{\mu}$-discrepancy.I show that, in the case of a quark model, the total$\Delta\alpha(M_Z^2)$-shift remains small. I also discuss the scenario where$\frac{1}{\pi}\Imm\Delta(t)$ is a constant in a finite $t$-region and show thatin this case, up to a $t_{\rm max}\precsim 5~\GeV^2$, the size of the total$\delta\Delta\alpha_{\rm had}^{(5)}(M_Z^2)$-shift, remains below or of theorder of the present error value on $\Delta\alpha_{\rm had}^{(5)}(M_Z^2)$.

On Constraints Between $\Delta\alpha_{\rm had}(M_Z^2)$ and $(g_{\mu}-2)_{\rm HVP}$