B below, validating our estimation of the mode linewidth $\hbar\Gamma_{\rm OSO}=80$ meV.

The dispersive modes of the system can be modeled by a dipolar Hamiltonian, where excitons in each valley are selectively coupled to degenerated OSO plasmonic modes of opposite wavevectors $\pm {\bf k}_{\rm SP}$, as depicted in Fig. 2 in the main text: $$\mathcal{H} = \sum_{k_x}\left[\mathcal{H}_{\tiny\textrm{OSO}}(k_x) +
  \mathcal{H}_{\tiny\textrm{ex}} +
  \mathcal{H}_{\tiny\textrm{int}}(k_x)\right], \label{eq:H}$$ which consists of three different contributions: $$\mathcal{H}_{\tiny\textrm{OSO}}(k_x) =
  \hbar\omega_{\tiny\textrm{OSO}}(k_x)\left(a^\dagger_{k_x}a_{k_x} +
    a^\dagger_{-k_x}a_{-k_x}\right),$$ $$\mathcal{H}_{\tiny\textrm{ex}}(k_x) =
  \hbar\omega_{\tiny\textrm{ex}}\left(b^\dagger_{K-k_x}b_{K-k_x} + b^\dagger_{K'+k_x}b_{K'+k_x}\right),$$ $$\begin{aligned}
  \mathcal{H}_{\tiny\textrm{int}}(k_x) &=& \hbar g\left(a^\dagger_{k_x}
      + a_{k_x}\right)\left(b^\dagger_{K'+k_x} + b_{K'+k_x}\right) \nonumber \\
   & & + \hbar g\left(a^\dagger_{-k_x}
      + a_{-k_x}\right)\left(b^\dagger_{K-k_x} + b_{K-k_x}\right),\end{aligned}$$ where $a(a^\dagger)$ are the lowering (raising) operators of the OSO plasmonic modes of energy $\hbar\omega_{\tiny\textrm{OSO}}(k_x)$, $b(b^\dagger)$ are the lowering (raising) operators of the exciton fields of energy $\hbar\omega_{\tiny\textrm{ex}}$, and $g = \Omega_R/2$ is the light-matter coupling frequency. In this hamiltonian the chiral light-chiral matter interaction is effectively accounted for by coupling excitons of the valley $K'(K)$ to plasmons propagating with wavevectors $k_x(-k_x)$. Moreover, the dispersion of the exciton energy can be neglected on the scale of the plasmonic wavevector ${\bf k}_{\rm SP}$.

Using the Hopfield procedure [@CiutiPRB2005], we can diagonalize the total Hamiltonian by finding polaritonic normal mode operators $P^\pm_{K(K')}$ associated with 