$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\M}{\mathcal{M}}$
Let $\M,\N$ be $d$-dimensional compact connected, oriented, diffeomorphic Riemannian manifolds (perhaps with boundary). Suppose that $d \ge 3$.
Is there a function $F:\mathbb{R}^+ \to \mathbb{R}^+$ such that the inequality
$$ \int_{\M} (\det df)^{\frac{2}{d}} \text{Vol}_{\M} \ge F\big(\,\int_{\M} \det df \, \text{Vol}_{\M} \, \,\big)=F\big(\text{Vol}(\N)\big)$$
holds for every orientaion-preserving diffeomorphism $f:\M \to \N$?
I allow $F$ to be any positive function. (It can depend on any properties of $\M,\N$).
*There are various equivalent ways to define the determinant of a linear map between oriented inner products spaces, here we can take as a definition
$$ \det df \cdot \text{Vol}_{\M}=f^*\text{Vol}_{\N},$$
So in particular $$ \text{Vol}(\N)=\int_{\N} \text{Vol}_{\N}=\int_{\M} f^*\text{Vol}_{\N}=\int_{\M} \det df \, \text{Vol}_{\M}.$$