Define $$S_{t}=\{(A,B)\in\mathbb{R}^{n\times n}\times\mathbb{R}^{n\times n}:\frac{\|A-B\|_F}{\sqrt{n}}\leq\sqrt{t}, \frac{\|A\|_F}{\sqrt{n}}\geq1-t, \frac{\|B\|_F}{\sqrt{n}}\geq1-t, \\\|A\|_{op}\leq 1, \|B\|_{op}\leq1\}$$ where $0<t<1$. So $S_t$ is a compact set in $\mathbb{R}^{n\times n}\times\mathbb{R}^{n\times n}$. Note that as $t\to0$, $S_t\to\{(A,A):A\in O(n)\}$.

Define a function over $\{(A,B)\in\mathbb{R}^{n\times n}\times\mathbb{R}^{n\times n}:\|A\|_{op}\leq 1, \|B\|_{op}\leq1\}$ as: $$f_{m,n}(A,B)=det(I-A^TA)^{\frac{m}{2}-\frac{n+1}{2}}det(I-B^TB)^{\frac{m}{2}-\frac{n+1}{2}}$$

Can we obtain the following conjecture? $$\frac{\int_{S_{2t}}f_{m,n}(A,B)}{\int_{S_{t}}f_{m,n}(A,B)}\leq C^{mn}$$ where $C$ is a constant independent of $m,n,t$, $m\geq n+1$. The integral is with respect to the Lebesgue measure on $\mathbb{R}^{n\times n}\times\mathbb{R}^{n\times n}$. It seems obvious since the function to be integrated is essentially a polynomial. However, I'm not able to formalize this. The tricky part is the integration region.