For the last decade, several probes have pointed to a cosmological tension between the amplitude of density fluctuations extrapolated from the cosmic microwave background within the standard cosmological model and the one encapsulated by the $S_8$ parameter from large scale structure. The origin of this $S_8$ tension has not yet been elucidated and may hint at systematics in the data, unaccounted effects from baryonic physics, or new physics beyond the standard model of cosmology. Baryonic physics may in principle provide a nonlinear solution to the tension by suppressing the matter power spectrum more strongly on nonlinear scales than is traditionally assumed. Such a solution would not worsen the Hubble tension, contrary to many other proposed solutions to the $S_8$ tension. However, no realistic baryonic feedback in hydrodynamical simulations provides the needed suppression as a function of redshift. Here, we point out that a scale-dependence of local-type primordial non-Gaussianities (PNG), with significant PNG at scales of a few Mpc, can provide the needed suppression, since such PNG can suppress the power spectrum at slightly larger scales than baryons do. We demonstrate this by devising collisionless numerical simulations of structure formation in boxes of 0.5 Gpc/h with scale-dependent local-type PNG. Our simple models show that, as a proof of principle, scale-dependent PNG, with a Gaussian random field for primodial density fluctuations on large scales and $f_{\rm NL} \simeq -300$ at $\lesssim 10$ Mpc scales, together with state-of-the-art baryonification of the matter power spectrum, can in principle solve the $S_8$ tension. The $S_8$ tension would then be a smoking-gun of non-trivial inflationary physics.