each valley, and obeying the following equation of motion at each $k_x$ $$\left[P^\pm_{K(K')},\mathcal{H}\right] = \hbar\omega_\pm
  P^\pm_{K(K')}, \label{eq:motion}$$ with $\omega_\pm>0$. In the rotating wave approximation (RWA), justified here by the moderate coupling strength (see below), $P^j_{\lambda}
\simeq \alpha^j_{\lambda}a +\beta^j_{\lambda}b$, $j\in\{+,-\}$ and $\lambda\in\{K,K'\}$. The plasmonic and excitonic Hopfield coefficients $\alpha^j_{\lambda}(k_x)$ and $\beta^j_{\lambda}(k_x)$ are obtained by diagonalizing the following matrix for every $k_x$ $$\left(
\begin{array}{cccc}
\hbar\omega_{\tiny\textrm{OSO}} & i\hbar\Omega_{\rm R} & 0 & 0 \\ 
-i\hbar\Omega_{\rm R} & \hbar\omega_{\tiny\textrm{ex}} & 0 & 0 \\ 
0 & 0 & -\hbar\omega_{\tiny\textrm{OSO}} & i\hbar\Omega_{\rm R}\\ 
0 & 0 & -i\hbar\Omega_{\rm R}& -\hbar\omega_{\tiny\textrm{ex}}
\end{array} 
\right). \label{eq:Matrix}$$

The dynamics of the coupled system will be ruled by the competition between the coherent evolution described by the Hamiltonian (\[eq:H\]) and the different dissipative processes contributing to the uncoupled modes linewidths. This can be taken into account by including the measured linewidths as imaginary parts in the excitonic and plasmonic mode energies (Weisskopf-Wigner approach). Under such conditions, we evaluate the eigenvalues $\omega_{\pm}$ of the the matrix (\[eq:Matrix\]). The real parts of $\omega_{\pm}$ are then fitted to the maxima of the angle resolved reflectivity maps presented on Fig. 3 in the main text, or to the zeros of the first derivative reflectivity 