Planar polyconvexity of exponentiated Hencky energies

For $F\in\mathbb{R}^{n\times n}$, define the exponentiated Hencky energy by

$$W_{\rm eH}(F)=\begin{cases}\displaystyle\frac{\mu}{k}e^{k\|\operatorname{dev}_n\log U\|^2}+\frac{\kappa}{2\widehat{k}}e^{\widehat{k}[(\log\det U)^2]}&\text{if }\det F>0,\\+\infty&\text{if }\det F\leq 0,\end{cases}$$

where $U$ is the right stretch tensor and $\mu,\kappa,k,\widehat{k}$ are parameters. Planar polyconvexity conjecture. The functions $W_{\rm eH}$ defined above are polyconvex for $n=2$, $\mu>0$, $\kappa>0$, $k\geq\frac14$, and $\widehat{k}\geq\frac18$. Polyconvexity implies Legendre–Hadamard ellipticity and can support existence results when suitable growth conditions hold; the source presents this as a conjectural extension of its planar rank-one-convexity theorem.