Note on fundamental physics tests from black hole imaging: Comment on "Hunting for extra dimensions in the shadow of Sagittarius A$^*$"

Several works over the past years have discussed the possibility of testing fundamental physics using Very Long Baseline Interferometry horizon-scale black hole (BH) images, such as the Event Horizon Telescope (EHT) images of M87$^*$ and Sagittarius A$^*$ (Sgr A$^*$), using the size $r_{\rm sh}$ and deviation from circularity $\Delta \mathcal{C}$ of the BH shadow. For the case of the EHT image of Sgr A$^*$, limits on $\Delta \mathcal{C}$ are not available due to the sparse interferometric coverage of the 2017 observations, alongside the short variability timescale of Sgr A$^*$ compared to M87$^*$. Concerning this point, we comment on the results of a recent preprint which purports to have derived new limits on extra dimensions using the deviation from circularity of Sgr A$^*$'s shadow. The latter is quoted to be $\lesssim 10\%$ as with M87$^*$, based on the "similarity" of the two shadows: however, this is an incorrect assumption, invalidating the subsequent results. In the immediate future, the simplest tests of fundamental physics from Sgr A$^*$'s image will therefore mostly have to rely on $r_{\rm sh}$, whereas additional observables such as the photon ring and azimuthal angle lapse should soon be available and allow for novel tests.