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. Is there any possible direction of literature that I can dive into to solve this problem?

This problem has been puzzling me for months. I tried using QR decomposition of $A$ and $B$ as a change of variable, but the term $\|A-B\|_F$ is hard to handle. Other terms $\|A\|_F, \|B\|_F, \|A\|_{op}, \|B\|_{op}$ are only related to the spectral property of the matrix, but $\|A-B\|_F$ is also related to the orientation of the matrix, which causes the main difficulty.