  For $t \ge 2$, let us call a digraph $D$ \emph{t-chordal} if all induced
directed cycles in $D$ have length equal to $t$. In a previous paper, we asked
for which $t$ it is true that $t$-chordal graphs with bounded clique number
have bounded dichromatic number. Recently, Aboulker, Bousquet, and de Verclos
answered this in the negative for $t=3$, that is, they gave a construction of
$3$-chordal digraphs with clique number at most $3$ and arbitrarily large
dichromatic number. In this paper, we extend their result, giving for each $t
\ge 3$ a construction of digraphs with clique number at most $3$ and
arbitrarily large dichromatic number, thus answering our question in the
negative. On the other hand, we show that a more restricted class, digraphs
with no induced directed cycle of length less than $t$, and no induced directed
$t$-vertex path, have bounded dichromatic number if their clique number is
bounded. We also show the following complexity result: for fixed $t \ge 2$, the
problem of determining whether a digraph is $t$-chordal is coNP-complete.
