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The converse is false. See the lecture notes by Chandan Singh Dalawat at http://arxiv.org/abs/math/0605326, which give some examples of varieties over finite extensions of $\mathbb{Q}_p$ whose $\ell$-adic cohomology is unramified for $\ell \ne p$ and crystalline at $\ell = p$, but the variety does not have good reduction.