Bhatt–Lurie’s obstruction-class conjecture for Frobenius liftings and the Hodge–Tate gerbe

Let $X$ be a smooth scheme over a field of characteristic $p$. Write $X_{\mathrm{fpqc}}$ for its fpqc site, let $T_{X/k}$ be its tangent bundle, and let $T_{X/k}^{\sharp}$ be the fpqc-group banding the Hodge–Tate gerbe $X^{\mathrm{HT}}$. Let $o_{F,\mathrm{fpqc}}\in \mathrm{H}^2(X_{\mathrm{fpqc}},\alpha_p\otimes T_X)$ be the obstruction to the existence of a Frobenius lifting, and let $o_{\mathrm{HT}}\in \mathrm{H}^2(X_{\mathrm{fpqc}},T^{\sharp}_{X/k})$ be the cohomology class of $X^{\mathrm{HT}}$. Bhatt–Lurie’s conjecture. The image of $o_{\mathrm{HT}}$ in $\mathrm{H}^2(X_{\mathrm{fpqc}},\alpha_p\otimes T_X)$, under the map induced by $\mathbb{G}_a^{\sharp}\to\alpha_p$ tensoring with $T_{X/k}$, is equal to $o_{F,\mathrm{fpqc}}$. This identifies the obstruction represented by the Hodge–Tate gerbe with the obstruction to lifting Frobenius, relating trivializations of the gerbe to Frobenius liftings; the paper presents it as a conjecture of Bhatt and Lurie.