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