  Thomassen conjectured that triangle-free planar graphs have exponentially
many 3-colorings. Recently, he disproved his conjecture by providing examples
of such graphs with $n$ vertices and at most $2^{15n/\log_2 n}$ 3-colorings. We
improve his construction, giving examples of such graphs with at most
$64^{n^{log_{9/2} 3}}<64^{n^{0.731}}$ 3-colorings. We conjecture this exponent
is optimal.
