Do there exist uncountably many abelian surfaces with good reduction over $\mathbb{Q}_p$ with pairwise non-isomorphic rational $p$-adic Tate modules?

If we took $l$-adic Tate modules there would be countably many at most by a result of Kisin and the fact that the moduli of ppav is of finite type.

This answer would suggest that there is only countably many $p$-adic Tate modules for elliptic curves with good reduction.

Is it at least true that there are uncountably many non-isomorphic crystalline representations of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ coming from geometry?

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