The p-integrality conjecture for the homological Chern character

Let $X$ be a variety, let $n$ be a positive integer, and let $p$ be a prime number. Write $\operatorname{ch}_{\dim X-n}[\mathcal{O}_X]$ for the $(\dim X-n)$-th component of the homological Chern character of the structure sheaf, and let $\mathbb{Z}_{(p)}$ denote the localization of $\mathbb{Z}$ at $p$. Then

$$p^{[n/(p-1)]}\cdot \operatorname{ch}_{\dim X-n}[\mathcal{O}_X] \in \operatorname{im}\bigl(\operatorname{CH}_{\dim X-n}(X)_{\mathbb{Z}_{(p)}} \to \operatorname{CH}_{\dim X-n}(X)_{\mathbb{Q}}\bigr).$$

The p-integrality conjecture. For every variety $X$, every positive integer $n$, and every prime number $p$, the displayed $p$-integrality assertion holds. The homological Chern character is generally defined with rational coefficients, so this predicts a precise bound on the denominators of its components for the structure sheaf. The statement concerns integrality in small codimension and is related to Steenrod operations and the Grothendieck–Riemann–Roch theorem; the supplied text does not indicate whether it has been resolved.