A multigraph G is an (s,q)-graph if every s-set of vertices in G supports at most q edges of G, counting multiplicities. Mubayi and Terry posed the problem of determining the maximum of the product of the edge-multiplicities in an (s,q)-graph on n vertices. We give an asymptotic solution to this problem for the family (s,q)=(2r,a(2r2)+ex(2r,Kr+1)−1) with r,a ∈ Z≥2. This greatly generalises previous results on the problem due to Mubayi and Terry and to Day, Treglown and the author, who between them had resolved the special case r=2. Our result asymptotically confirms an infinite family of cases in (and overcomes a major obstacle to a resolution of) a conjecture of Day, Treglown and the author.