Adapted metric conjecture for p-sectional hyperbolic sets

Let $X$ be a $C^1$ vector field with flow $X_t$, and let $\Gamma$ be a $p$-sectional hyperbolic set with invariant splitting $T_\Gamma M=E\oplus F$. A metric is required to provide constants $\mu>0$ such that, for every $x\in\Gamma$ and $t>0$, the stable bundle contracts, domination holds, and every $p$-dimensional subspace $L_x\subset F_x$ expands. Adapted metric conjecture. There exists a metric such that, for some constant $\mu>0$ and all $t>0$,

$$|DX_t\mid_{E_x}|\le e^{-\mu t},$$ $$|DX_t\mid_{E_x}|\cdot\big|(DX_t\mid_{F_x})^{-1}\big|\le e^{-\mu t},$$

and

$$|\wedge^p DX_t(x)\vert_{L_x}|>e^{\mu t}$$

for every $p$-dimensional linear subspace $L_x\subset F_x$. The conjecture proposes that the adapted-metric conclusion previously obtained for sectional hyperbolic sets extends to the more general $p$-sectional setting; no resolution is supplied here.