Syntomic sheaf conjecture for quasisyntomic semiperfectoid rings

Let $p$ be a prime, let $i$ be an integer, and let $\mathrm{QRSPerfd}$ denote the site of quasisyntomic semiperfectoid rings. Write $\mathbb Z_p(i)$ for the corresponding syntomic sheaf of complexes.

Syntomic sheaf conjecture. The sheaf of complexes $\mathbb Z_p(i)$ on $\mathrm{QRSPerfd}$ is locally concentrated in degree $0$ and is given by a $p$-torsion-free sheaf. Equivalently, on $\mathrm{QRSPerfd}$ the sheafification of $K_{2i}(-;\mathbb Z_p)$ is $p$-torsion-free and the sheafification of $K_{2i-1}(-;\mathbb Z_p)$ vanishes; this vanishing is equivalent to the surjectivity of

$$\varphi_i-1:\mathcal N^{\geq i}\widehat{\mathbbl{\Delta}}_{(-)}\{i\}\longrightarrow \widehat{\mathbbl{\Delta}}_{(-)}\{i\}.$$

The preceding proposition proves the case $n=1$, while the source indicates that the conjecture is proved in characteristic $p$. The general integral statement is therefore recorded here as resolved according to the supplied status evidence.