Integrality conjecture for the Chern character

Let $p$ be a prime number and $n$ an integer. Let $S$ be a regular scheme, $X$ a scheme of finite type over $S$, $d$ an integer, and $x\in K_0'(X)_{(d)}$. For each integer $i\leq n$, let $v_p(\operatorname{ch}_{d-i}x)$ denote the $p$-adic valuation of the $(d-i)$-th component of the Chern character. Integrality conjecture. We have, for all $i\leq n$,

$$v_p(\operatorname{ch}_{d-i}x)\geq-\left\lfloor\frac{i}{p-1}\right\rfloor.$$

This conjecture concerns the denominators occurring in the Chern character and predicts a uniform lower bound for their $p$-adic valuations. Its formulation depends on the prime $p$ and the integer $n$; the supplied source does not establish its resolution.