Unstoppable brane-flux decay of $\overline{\text{D6}}$ branes

We investigate $p$ $\overline{\text{D6}}$ branes inside a flux throat that carries $K \times M$ D6 charges with $K$ the 3-form flux quantum and $M$ the Romans mass. We find that within the calculable supergravity regime where $g_s p$ is large, the $\overline{\text{D6}}$ branes annihilate immediately against the fluxes despite the existence of a metastable state at small $p/M$ in the probe approximation. The crucial property that causes this naive conflict with effective field theory is a singularity in the 3-form flux, which we cut off at string scale. Our result explains the absence of regular solutions at finite temperature and suggests there should be a smooth time-dependent solution. We also discuss the qualitative differences between $\overline{\text{D6}}$ branes and $\overline{\text{D3}}$ branes, which makes it a priori not obvious to conclude the same instability for $\overline{\text{D3}}$ branes.