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In a paper published in 2023, Wagner introduced and studied Jacobi forms withcomplex multiplication, and gave several applications. One such application wasin 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 theRamanujan-type congruences satisfied by $\overline{p}_{k,j}$. In this note, wegive an asymptotic formula for the number of $(k,j)$-coloured overpartitionsand prove that any crank constructed by Wagner is asymptoticallyequidistributed on arithmetic progressions, following several recent papers inthe literature.

Crank equidistribution and $(k,j)$-overlined partitions