  Using the supergravity dual and the plane-wave limit as a guide, we
conjecture the asymptotic large coupling form of the Hagedorn temperature for
planar $\mathcal{N}=4$ super Yang-Mills to order $1/\sqrt{\lambda}$. This is
two orders beyond the presently known behavior. Using the quantum spectral
curve procedure of Harmark and Wilhelm, we show that our conjectured form is in
excellent agreement with the numerical results.
