Rank-one convexity region for the isochoric exponentiated Hencky energy

Let $U$ be the right stretch tensor, let $\operatorname{dev}_n\log U$ be the deviatoric part of its logarithm, and define

$$\mathcal{E}(W_{\rm eH},\operatorname{LH},U,27)=\{U\in\operatorname{PSym}(3)\mid\|\operatorname{dev}_3\log U\|^2<27\}.$$

Rank-one convexity region conjecture. For $n=2,3$, the energy $F\mapsto\frac{\mu}{k}e^{k\|\operatorname{dev}_n\log U\|^2}$ with $k>\frac{3}{16}$ is rank-one convex on a set containing the large cone $\mathcal{E}(W_{\rm eH},\operatorname{LH},U,27)$. The source describes this as a conjecture arising from unsuccessful attempts to establish the global ellipticity properties.