maps shown here in Fig. \[fig:S1\] (a) and (b). Both procedures give the same best fit that yields the polaritonic branch splitting as [@Houdre2005; @TormaBarnes2015] $$\hbar(\omega_+-\omega_-) = 2\sqrt{\left(\hbar\Omega_R\right)^2 - \left( \hbar\gamma_{\tiny\textrm{ex}}- \hbar\Gamma_{\tiny\textrm{OSO}}\right)^2}
\label{eq:SCsplit}$$ which equals $40$ meV. From the determination (see above) of the FWHM of the excitonic $\hbar\gamma_{\tiny\textrm{ex}}$ and plasmonic $\hbar\Gamma_{\tiny\textrm{OSO}}$ modes, we evaluate a Rabi energy $ \hbar\Omega_R=70$ meV.

These values give a ratio $$\left(\hbar\Omega_R\right)^2 / \left(\left(\hbar\gamma_{\tiny\textrm{ex}}\right)^2 +
 \left(\hbar  \Gamma_{\tiny\textrm{OSO}}\right)^2\right) = 0.69,$$ above the $0.5$ threshold which is the strong coupling criterion -see [@Houdre2005] for a detailed discussion. This figure-of-merit of $0.69>0.5$ therefore clearly demonstrates that our system is operating in the strong coupling regime.

Interestingly, the intervalley scattering rate $\hbar\gamma_{KK'}$ does not enter in this strong coupling criterion. Indeed, such events corresponding to an inversion of the valley indices $K\leftrightarrow K'$ do not contribute to the measured excitonic linewidth, and are thus not detrimental to the observation of strong coupling. In the $\hbar\Omega_R \ll
\hbar\gamma_{KK'}$ limit, the Hamiltonian (\[eq:H\]) would reduce to the usual RWA Hamiltonian and the valley contrasting chiralitonic behavior would be lost. The results gathered in Fig. 4 in the main text clearly show that this is not the case for our system, allowing 