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We investigate the sums (1/√(H)) ∑_X < n ≤ X+Hχ(n), where χ is a fixed non-principal Dirichlet character modulo a prime q, and 0 ≤ X ≤ q-1 is uniformly random. Davenport and Erdős, and more recently Lamzouri, proved central limit theorems for these sums provided H →∞ and (log H)/log q → 0 as q →∞, and Lamzouri conjectured these should hold subject to the much weaker upper bound H=o(q/log q). We prove this is false for some χ, even when H = q/log^Aq for any fixed A > 0. On the other hand, wearXiv:2203.09448v1 fatcat:c2yni4aa65d4jgqzloaruakvce