= Solution
Assume (v). If the generic fibre $M=K\otimes_{\mathcal O}W$ had a nonzero proper $KG$-submodule, intersecting it with $W$ and rescaling to obtain a saturated lattice would give a nonzero proper $kG$-submodule of $\overline W$. Thus $M$ is simple.
Because $\overline W$ is projective, its restriction to a Sylow p-subgroup $S$ is projective. The local algebra $kS$ has only free finitely generated projectives, so $|S|=p^d$ divides
$$
\dim_k\overline W=\operatorname{rank}_{\mathcal O}W=\dim_KM=n.
$$
Consequently
$$
\boxed{\text{(v)}\Longrightarrow\text{(i)}.}
$$
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