For $\mathbf x=(x_1,\ldots, x_n), \mathbf y=(y_1,\ldots, y_n) \in \mathbb R^n$, I am interested in the function $$F(\mathbf x, \mathbf y) = \frac{\prod_{i<j}(\sinh|x_i-x_j| \sinh|y_i-y_j|)}{\prod_{1\leq i,j\leq n} \sinh |x_i-y_j|}.$$
Note that $F$ is divergent if $x_i=y_j$ for some $i,j$. Hence, we fix $\delta>0$ and restrict the domain of $F$ to $|x_i-y_j|>\delta$ for all $1\leq i,j\leq n$.
Question: What is the upper bound of $F$ on this restricted domain? (Furthermore, it would be better if one has some decay estimates on $F$.)
Remarks.
We may assume that $x_1\leq\cdots \leq x_n$ and $y_1\leq \cdots \leq y_n$.
I think $F$ should be bounded by the following heuristic argument. We have to consider the behavior of $F$ for large $x_i$ and $y_j$. Then, approximating $\sinh |x| \approx e^{|x|}$, we obtain $$F(\mathbf x, \mathbf y) \approx \exp\left( \sum_{i<j} (|x_i-x_j|+|y_i-y_j|) - \sum_{i,j} |x_i-y_j|\right).$$ However, by my previous question $2\sum_{i,j} |x_i-y_j| \geq \sum_{i,j} (|x_i-x_j| + |y_i-y_j|)$?, The RHS is $\leq 1$.
Actually, examining the results in my previous question, we have at least $$\sum_{i<j} (|x_i-x_j|+|y_i-y_j|) - \sum_{i,j} |x_i-y_j| \leq -\sum_i |x_i-y_i|.$$ Hence, one may expect the exponential decay estimate on $F$, which is much better than $\leq 1$ estimate. However, I did not find a rigorous way to justify on the part $\sinh |x| \approx e^{|x|}$.
Any help on this question (finding $\sup F$, or finding rigorous decay estimates) would be appreciated.