  Contrary to the expectation arising from the tanglegram Kuratowski theorem of
\'E. Czabarka, L.A. Sz\'ekely and S. Wagner [SIAM J. Discrete Math. 31(3):
1732--1750, (2017)], we construct an infinite antichain of planar tanglegrams
with respect to the induced subtanglegram partial order. R.E. Tarjan, R. Laver,
D.A. Spielman and M. B\'ona, and possibly others, showed that the partially
ordered set of finite permutations ordered by deletion of entries contains an
infinite antichain, i.e. there exists an infinite collection of permutations,
such that none of them contains another as a pattern. Our construction adds a
twist to the construction of Spielman and B\'ona [Electr. J. Comb, Vol. 7. N2.]
