and $\bar\varepsilon_{3/2}$, we will work with the central values $\delta_{\rm EW}=0.64$ and $\bar\varepsilon_{3/2}=0.21$. By the time the experimental data will be sufficiently precise to perform the New Physics searches proposed in this work, the errors on these parameters are likely to be reduced by a significant amount.

Let us first discuss the case where New Physics induces arbitrary CP-violating contributions to the $B\to\pi K$ decay amplitudes, while preserving isospin symmetry. Then the only change with respect to the SM would be that the weak phase $\varphi$ may no longer be negligible. We obtain $$\begin{aligned}
\label{Rstphi}
   R_*^{-1} &=& 1 + 2\bar\varepsilon_{3/2} \Big[
    \cos\varphi\,(\delta_{\rm EW}-\cos\gamma)\cos\phi
    + \sin\varphi\sin\gamma\cos(\phi-\phi_\rho) \Big] \nonumber\\
   &&\mbox{}+ \bar\varepsilon_{3/2}^2 
    (1 - 2\delta_{\rm EW}\cos\gamma + \delta_{\rm EW}^2) 
    \nonumber\\
   &\le& 1 + 2\bar\varepsilon_{3/2} \Big[ 
    \cos\varphi\,|\delta_{\rm EW}-\cos\gamma|
    + |\sin\varphi\sin\gamma| \Big] \nonumber\\
   &&\mbox{}+ \bar\varepsilon_{3/2}^2 
    (1 - 2\delta_{\rm EW}\cos\gamma + \delta_{\rm EW}^2) 
    \nonumber\\
   &\le& \left( 1 + \bar\varepsilon_{3/2}
   \sqrt{1 - 2\delta_{\rm EW}\cos\gamma + \delta_{\rm EW}^2}
   \right)^2 \,.\end{aligned}$$ In deriving the upper bounds we have varied the strong-interaction phases $\phi$ and $\phi_\rho$ independently. Analogous lower bounds are obtained as previously by changing the sign of $\bar\varepsilon_{3/2}$. The last inequality in (\[Rstphi\]) is remarkable in that it holds for arbitrary isospin-conserving New Physics effects no matter how 