  A graph is cubical if it is a subgraph of a hypercube. For a cubical graph
$H$ and a hypercube $Q_n$, $ex(Q_n, H)$ is the largest number of edges in an
$H$-free subgraph of $Q_n$. If $ex(Q_n, H)$ is at least a positive proportion
of the number of edges in $Q_n$, $H$ is said to have a positive Tur\'an density
in a hypercube or simply a positive Tur\'an density; otherwise it has a zero
Tur\'an density. Determining $ex(Q_n, H)$ and even identifying whether $H$ has
a positive or a zero Tur\'an density remains a widely open question for general
$H$. By relating extremal numbers in a hypercube and certain corresponding
hypergraphs, Conlon found a large class of cubical graphs, ones having
so-called partite representation, that have a zero Tur\'an density. He raised a
question whether this gives a characterisation, i.e., whether a cubical graph
has zero Tur\'an density if and only if it has partite representation. Here, we
show that, as suspected by Conlon, this is not the case. We give an example of
a class of cubical graphs which have no partite representation, but on the
other hand, have a zero Tur\'an density. In addition, we show that any graph
whose every block has partite representation has a zero Tur\'an density in a
hypercube.
