Breuil-Kisin Modules via crystalline cohomology

For a perfect field $k$ of characteristic $p>0$ and a smooth and proper formal scheme $\mathcal{X}$ over the ring of integers of a finite and totally ramified extension $K$ of $W(k)[1\mathrm{/}p]$, we propose a cohomological construction of the Breuil-Kisin modules attached to the $p$-adic 'etale cohomology $H_{\acute{\mathrm{e}}\mathrm{t}}^i({\mathcal{X}}_{\bar{K}},{\mathbf{Z}}_p)$. We then prove that our proposal works when $p>2$, $i<p-1$, and the crystalline cohomology of the special fiber of $\mathcal{X}$ is torsion-free in degrees $i$ and $i+1$.