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Motivated by a question of Hansen and Li, we show that a smooth and properrigid analytic space $X$ with projective reduction satisfies Hodge symmetry inthe following situations: (1) the base non-archimedean field $K$ is of residuecharacteristic zero, (2) $K$ is $p$-adic and $X$ has good ordinary reduction,(3) $K$ is $p$-adic and $X$ has "combinatorial reduction."' We also reprove aversion of their result, Hodge symmetry for $H^1$, without the use of modulispaces of semistable sheaves. All of this relies on cases of Kato's log hardLefschetz conjecture, which we prove for $H^1$ and for log schemes of"combinatorial type."

Hodge symmetry for rigid varieties via log hard Lefschetz