  We introduce two classes of morphisms over the alphabet $A=\{0,1\}$ whose
fixed points contain infinitely many antipalindromic factors. An antipalindrome
is a finite word invariant under the action of the antimorphism
$\mathrm{E}:\{0,1\}^*\to\{0,1\}^*$, defined by $\mathrm{E}(w_1\cdots
w_n)=(1-w_{n})\cdots(1-w_1)$. We conjecture that these two classes contain all
morphisms (up to conjugation) which generate infinite words with infinitely
many antipalindromes. This is an analogue to the famous HKS conjecture
concerning infinite words containing infinitely many palindromes. We prove our
conjecture for two special classes of morphisms, namely (i) uniform morphisms
and (ii) morphisms with fixed points containing also infinitely many
palindromes.
