= Solution
Let $L$ be any <distributive lattice>. Its <Stone map of a distributive lattice>
$$
a\longmapsto a^*
$$
is an injective lattice homomorphism from $L$ into the lattice of <clopen up-set>[clopen up-sets] of its <Priestley dual space>. In particular these images are open subsets of the underlying <topological space>, and the map preserves $\bot$, $\top$, finite meets and finite joins.
Now assume that an implication-free formula $\phi$ is valid under every lattice valuation in every topological space. Given any valuation of its variables in any distributive lattice $L$, compose it with the Stone map. Topological validity says that the resulting value of $\phi$ is the whole Priestley space. Injectivity of the Stone map then says that the original value of $\phi$ was $1_L$. Hence $\phi$ is valid in every distributive lattice.
By <completeness of implication-free intuitionistic propositional logic for distributive lattices>, $\phi$ is provable in the <implication-free fragment of intuitionistic propositional logic>.
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