We consider the harmonic map heat flow from $\mathbb{R}^d$ into $\mathbb{S}^d$, which is energy-supercritical for $d \geq 3$. It is well known that for $3 \leq d \leq 6$ the flow admits finite-time blowup via self-similar solutions, so-called shrinkers, within the class of corotational maps. Numerical experiments suggest that one of these, the ground state, is stable and thus describes the generic blowup behaviour of corotational solutions. In this talk, I present results on the existence and nonlinear asymptotic stability of the ground state in all dimensions $3 \leq d \leq 6$, obtained via computer-assisted proofs.