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We construct a counterexample for an analogue of Masur's criterion in thesetting of Teichm\"uller space equipped with the Thurston metric. For that, wefind a minimal, filling, non-uniquely ergodic lamination $\lambda$ on theseven-times punctured sphere with uniformly bounded annular projectiondistances. Then we show that a geodesic in the corresponding Teichm\"ullerspace that converges to $\lambda$, stays in the thick part for the whole time.

Masur's criterion does not hold in the Thurston metric