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the variation of the radiative decay rates of emitters in a dielectric nano-object placed on a glass surface [19]. It turns out that for the parameters (indices of refraction, particle size) of relevance to our measurements, τ could vary within a factor of three, depending on the orientation of each emitter and its distance from the interface. 

The details of the inhomogeneities in τ are not important to our current work because we will be only interested in its modification when the scattering medium is added. In fact, in what follows we will only investigate the changes of the long-lifetime component of the measurements. As explained below, while several systematic effects could cause shortening of the fluorescence lifetime, its increase provides a robust evidence for the modification of the density of states in the scattering medium. Fig. 3b displays the histogram of this component for DNCs placed on a glass cover slide in air. The variation from 8 ns to 37 ns is in agreement with previous reports [11, 20]. We checked the validity of this attribution by measuring the fluorescence lifetime τ w of the same DNCs after adding water on top of the sample (see Fig. 1c). The change in the radiative lifetime of an emitter is inversely proportional to the refractive index of the environment so we expect τ to be reduced under water. Figure 3c shows a histogram of the resulting values normalized to the lifetimes of each DNC in air. The mean value of τ w /τ a amounts to 0.9, where τ a denotes the fluorescence lifetime at the air-glass interface. This outcome is in very good agreement with the expected value of 0.88 if one considers the refractive index of the surrounding environment to be the average of the refractive indices of the media in the lower and upper halves of the glass-air or glass- water interfaces. Note that we do not have to account for local field effects since the NV centers embedded in the diamond matrix do not experience a change of environment at the 

