  According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic,
compact and connected subset of the group $\operatorname{GL}^+(2)$ of
invertible $2\times2-\,$matrices is rank-one convex if and only if it is
polyconvex. In a 2005 Journal of Convex Analysis article by Alexander~Mielke,
it has been conjectured that the equivalence of rank-one convexity and
polyconvexity holds for isotropic functions on $\operatorname{GL}^+(2)$ as
well, provided their sublevel sets satisfy the corresponding requirements. We
negatively answer this conjecture by giving an explicit example of a function
$W:\operatorname{GL}^+\to\mathbb{R}$ which is not polyconvex, but rank-one
convex as well as isotropic with compact and connected sublevel sets.
