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We prove that for every integer $t$ there exists a constant $c(t)$ such that every $n$-vertex even-hole-free graph with no clique of size $t$ and no induced subgraph isomorphic to a generalized $t$-pyramid has treewidth at most $c(t)\\log{n}$. This settles a special case of a conjecture of Sintiari and Trotignon; this bound is also best possible for the class. 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