We study the query complexity of finding a Tarski fixed point over $[n]^k$. Previous work has left a large gap between $\smash{{\Omega}(\log^2 n)}$ and $\smash{\log^{O(k)}n}$. We show that both of the previous upper and lower bounds were far from tight: for every $k\geq 3$,
$$
\Omega\left((\log n)^{\frac{1}{2}\lceil \log k\rceil}\right)\le
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