Relative crystalline representations and $p$-divisible groups in the small ramification case

Let $k$ be a perfect field of characteristic $p>2$, and let $K$ be a finite totally ramified extension over $W(k)[\frac{1}{p}]$ of ramification degree $e$. Let $R_0$ be a relative base ring over $W(k)\langle t_1^{\pm 1}, \ldots, t_m^{\pm 1}\rangle$satisfyingsomemildconditions,andlet$R = R_0\otimes_{W(k)}$\mathcal{O}$_K$\mathrm{.}Weshowthatif$e < p-1$, then every crystalline representation of ${\pi }_1^{\acute{\text{e}}\text{t}}(\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}R[\frac{1}{p}])$ with Hodge-Tate weights in $[0,1]$ arises from a $p$-divisible group over $R$.