The main torsion-length inequality for crystalline and de Rham cohomology

Let $X/\mathcal{O}_K$ be the fixed smooth proper scheme in the paper, with special fibre $X_k$, where $W$ is the ring of Witt vectors and $e$ is the ramification index of $\mathcal{O}_K$ over its maximal unramified subring. For each $i\geqslant 0$, write

$$\ell^i_\mathrm{crys}=\ell_W\left(\mathrm{H}^i_\mathrm{crys}(X_k/W)[p^\infty]\right),\qquad \ell^i_\mathrm{dR}=\ell_{\mathcal{O}_K}\left(\mathrm{H}^i_\mathrm{dR}(X/\mathcal{O}_K)[p^\infty]\right).$$

Main conjecture. For all $i\geqslant 0$,

$$\ell^i_\mathrm{crys}\leqslant \ell^i_\mathrm{dR}\leqslant e\cdot\ell^i_\mathrm{crys}.$$

These inequalities would give a uniform comparison between the lengths of crystalline and de Rham torsion. The paper develops an integral crystalline–de Rham comparison framework and verifies the conjecture in a small number of cases, while the general assertion remains open.