The asymptotic form of the Hagedorn temperature in planar $\mathcal{N}=4$ super Yang-Mills

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 $1\mathrm{/}\sqrt{\lambda }$. 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.