Some dwarf galaxies are within the Mondian regime at all radii, i.e., the gravitational acceleration provided by the observed baryons is always below the threshold of $g_†\simeq 1.2\times 10^{-10}\,{\rm m\,s^{-2}}$. These dwarf galaxies often show cores, in the sense that assuming Newton's gravity to explain their rotation curves, the total density profile $\rho(r)$ presents a central plateau or {\em core} ($d\log \rho/d\log r\rightarrow 0$ when $r\rightarrow 0$). Here we show that under MOND gravity, the existence of this core implies a baryon content whose density $g_{\rm bar}$ must decrease toward the center of the gravitational potential ($g_{\rm bar}\rightarrow 0$ when $r\rightarrow 0$). Such drop of baryons toward the central region is neither observed nor appears in numerical simulations of galaxy formation following MOND gravity. We analyze the problem posed for MOND as well as possible workarounds.