In 1951, Higman constructed a remarkable group
$$H=\left\langle a,b,c,d \, \left| \, b^a = b^2, c^b = c^2, d^c =
d^2, a^d = a^2 \right. \right\rangle$$
and used it to produce the first examples of infinite simple groups. By studying fixed points of certain finite state transducers, we show the conjugacy problem in $H$ is decidable (for all inputs).

Diekert, Laun & Ushakov have recently shown the word problem in $H$ is solvable in polynomial time, using the power circuit technology of Myasnikov, Ushakov & Won.
Building on this work, we show in a strongly generic setting that the conjugacy problem has a polynomial time solution.