safely neglected. As we explain in the main text, the resulting dispersion spectra are broadened by the contribution of different plasmonic modes as well as coupled and uncoupled exciton populations.

In order to highlight the polaritonic contribution in the absorption spectra, we calculate the first derivative of the reflectivity dispersions $\textrm{d} \left[
R_{\tiny\textrm{sample}}/R_{\tiny\textrm{substrate}} \right] / \textrm{d}E$. The derivative was approximated by interpolating the reflectivity spectra on an equally spaced wavelength grid of step $\Delta \lambda = 0.55$ nm and using the following finite difference expression valid up to fourth order in the grid step: $$\frac{\textrm{d}R}{\textrm{d}\lambda}(\lambda_0) \simeq \frac{\frac{1}{12}R(\lambda_{-2})
    -\frac{2}{3}R(\lambda_{-1}) + \frac{2}{3}R(\lambda_{+1}) -
    \frac{1}{12}R(\lambda_{+2})}{\Delta \lambda},$$ where $R(\lambda_{n})$ is the reflectivity evaluated $n$ steps away from $\lambda_{0}$. The resulting first derivative reflectivity spectra were then converted to an energy scale and plotted as dispersion diagrams in Fig. \[fig:S1\] (a) and (b).

In these first derivative reflectivity maps, the zero-crossing points correspond to the peak positions of the modes, and the maxima and minima indicate the inflection points of the reflectivity lineshapes. At the excitonic asymptotes of the dispersion curves, where the polaritonic linewidth is expected to match that of the bare WS$_2$ exciton, we read a linewidth of $26$ meV from the maximum 