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We prove that rank-$(n-1)$ convexity does not imply ${\mathcalS}$-quasiconvexity (i.e., quasiconvexity with respect to divergence freefields) in ${\mathbb M}^{m\times n}$ for $m>n$, by adapting the well-knownSverak's counterexample [5] to the solenoidal setting. On the other hand, wealso remark that rank-$(n-1)$ convexity and ${\mathcal S}$-quasiconvexity turnout to be equivalent in the space of $n\times n$ diagonal matrices. Thisfollows by a generalization of Mueller's work [4].

On the relationship between rank-$(n-1)$ convexity and ${\mathcal S}$-quasiconvexity