Nonexistence of globally TSTS-M$^+$-monotone isochoric energies

Let $U$ be the right stretch tensor and let $K_2^2=\|\operatorname{dev}_n\log U\|^2$. Global TSTS-M$^+$ nonexistence conjecture. For $n=2,3$, there is no elastic energy of the form

$$W=W(K_2^2)=W(\|\operatorname{dev}_n\log U\|^2)$$

such that $F\mapsto W(\|\operatorname{dev}_n\log U\|^2)$ satisfies the TSTS-M$^+$ condition in $\operatorname{GL}^+(n)$ over the entire deformation range. The claim concerns the impossibility of global strict stress–strain monotonicity for energies depending only on the deviatoric logarithmic strain invariant.