Hopkins–Smith vanishing conjecture for the Adams–Novikov vanishing curve

Let $g_{\mathrm{BP}}(n)$ be the $E_\infty$ vanishing curve of the Adams–Novikov spectral sequence, defined by

$$g_{\mathrm{BP}}(n)=\max\{s\mid {}^{\mathrm{AN}}E_\infty^{s,k+s}\neq0\text{ for some }k\leq n\}.$$

Hopkins–Smith vanishing conjecture. The vanishing curve satisfies

$$g_{\mathrm{BP}}(n)=n^{\frac12+o(1)}.$$

A strong form of the nilpotence theorem gives sublinear growth, while this conjecture predicts the leading-order asymptotics; the supplied text does not state a resolution.