  Daligault, Rao and Thomass\'e asked whether a hereditary class of graphs
well-quasi-ordered by the induced subgraph relation has bounded clique-width.
Lozin, Razgon and Zamaraev recently showed that this is not true for classes
defined by infinitely many forbidden induced subgraphs. However, in the case of
finitely many forbidden induced subgraphs the question remains open and we
conjecture that in this case the answer is positive. The conjecture is known to
hold for classes of graphs defined by a single forbidden induced subgraph $H$,
as such graphs are well-quasi-ordered and are of bounded clique-width if and
only if $H$ is an induced subgraph of $P_4$. For bigenic classes of graphs,
i.e. ones defined by two forbidden induced subgraphs, there are several open
cases in both classifications. In the present paper we obtain a number of new
results on well-quasi-orderability of bigenic classes, each of which supports
the conjecture.
