Can supernovae quench star formation in high-$z$ galaxies?

JWST is providing the unique opportunity to directly study feedback processes regulating star formation (SF) in early galaxies. The two $z>5$ quiescent systems (JADES-GS-z7-01-QU and MACS0417-z5BBG) detected so far show a recent starburst after which SF is suppressed. To clarify whether such quenching is due to supernova (SN) feedback, we have developed a minimal physical model. We derive a condition on the minimum star formation rate, $\rm SFR_{min}$, lasting for a time interval $\Delta t_{b}$, required to quench SF in a galaxy at redshift $z$, with gas metallicity $Z$, and hosted by a halo of mass $M_h$. We find that lower $(z, Z, M_h)$ systems are more easily quenched. We then apply the condition to JADES-GS-z7-01-QU ($z=7.3$, $M_\star=10^{8.6} M_\odot$) and MACS0417-z5BBG ($z=5.2$, $M_\star=10^{7.6} M_\odot$), and find that SN feedback largely fails to reproduce the observed quenched SF history. Alternatively, we suggest that SF is rapidly suppressed by radiation-driven dusty outflows sustained by the high specific SFR (43 and 25 Gyr$^{-1}$, respectively) of the two galaxies. Our model provides a simple tool to interpret the SF histories of post-starburst galaxies, and unravel quenching mechanisms from incoming JWST data.