  In a paper published in 2023, Wagner introduced and studied Jacobi forms with
complex multiplication, and gave several applications. One such application was
in constructing a new doubly-infinite family of partition-theoretic objects,
called $(k,j)$-coloured overpartitions and labelled by $\overline{p}_{k,j}$,
and using the Jacobi forms to construct crank functions which explain the
Ramanujan-type congruences satisfied by $\overline{p}_{k,j}$. In this note, we
give an asymptotic formula for the number of $(k,j)$-coloured overpartitions
and prove that any crank constructed by Wagner is asymptotically
equidistributed on arithmetic progressions, following several recent papers in
the literature.
