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

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