Dwork's conjecture on asymptotic growth of horizontal sections

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with non-trivial valuation, let $M$ be a finite free differential module over $K[\![t]\!]_0$, and let

$$H^0(M\otimes K\{t\})$$

be its space of horizontal sections. Write $n=\dim_K H^0(M\otimes K\{t\})$, and let $\mathrm{Fil}_{\delta}H^0(M\otimes K\{t\})$ be the filtration defined using the coefficient-growth filtration $K[\![t]\!]_{\delta}$. Dwork's conjecture. One has

$$\mathrm{Fil}_{n-1}H^0(M\otimes K\{t\})=H^0(M\otimes K\{t\}).$$

This generalizes Dwork's theorem from solvable differential modules, where the bound is given by the rank, to modules that need not be solvable.