  The \emph{Ramsey multiplicity constant} of a graph $H$ is the limit as $n$
tends to infinity of the minimum density of monochromatic labeled copies of $H$
in a $2$-edge colouring of $K_n$. Fox and Wigderson recently identified a large
family of graphs whose Ramsey multiplicity constants are attained by sequences
of ``Tur\'an colourings''; i.e. colourings in which one of the colour classes
forms the edge set of a balanced complete multipartite graph. Each graph in
their family comes from taking a connected non-3-colourable graph with a
critical edge and adding many pendant edges. We extend their result to an
off-diagonal variant of the Ramsey multiplicity constant which involves
minimizing a weighted sum of red copies of one graph and blue copies of
another.
