Higher-dimensional Lyapunov-exponent conjecture for diffeomorphisms

Let $M$ be a compact manifold of any dimension, let $f:M\circlearrowleft$ be a $C^\infty$ diffeomorphism, and for $k$ define

$$\Sigma^k\chi(x):=\limsup_n\frac{1}{n}\lVert\Lambda^k d_xf^n\rVert,$$

where $\Lambda^k df$ is the action induced by $f$ on the $k$th exterior power of $TM$, with $\Sigma^0\chi=0$. Higher-dimensional Lyapunov-exponent conjecture. If

$$\operatorname{Leb}\left(\Sigma^k\chi>\Sigma^{k-1}\chi\geq 0\right)>0,$$

then there exists an ergodic measure with at least $k$ positive Lyapunov exponents whose entropy is greater than or equal to the sum of its $k$ smallest positive Lyapunov exponents.

This proposes a higher-dimensional analogue of the surface result, relating positive-volume expansion in successive exterior powers to an invariant measure with several positive exponents and a corresponding entropy lower bound. The source presents it as a conjecture and gives no resolution.