# Modulus of image of a curve family in a rectangle

I don't expect to get a positive answer to this question but I may as well try.

Let $$R$$ be the rectangle in $$\mathbb{C}$$ given by $$\{z=x+iy: 0\leq x \leq l, 0 \leq y \leq h\}$$ for some $$l,h>0$$. Let $$\Gamma$$ denote the set of a curves connecting the top and bottom. That is, the set of curves connecting $$[0,l]\times \{0\}$$ and $$[0,l]\times \{ih\}$$. Let $$\Gamma'$$ be the curves connecting the left and right sides. It is well-known that the modulus of $$\Gamma$$, $$\Gamma'$$ are $$l/h$$, $$h/l$$ respectively.

Consider a sequence of quasiconformal maps $$f_n$$ with $$K(f_n)\to \infty$$. We obtain curve families $$\Gamma_n:=f_n(\Gamma)$$, $$\Gamma_n':=f_n(\Gamma')$$ in $$f_n(R)$$. Suppose we know $$\textrm{mod}(\Gamma_n)\to \infty$$ as $$n\to \infty$$ and we also have upper and lower bounds on the lengths of the curves in $$\Gamma_n$$ and $$\Gamma_n'$$. Then can we say anything precise about the rate of growth of $$\textrm{mod}(\Gamma_n)$$? In particular I am looking for a comparison between $$\textrm{mod}(\Gamma_n)$$ and $$\textrm{mod}(\Gamma)$$, namely an inequality of the form $$\textrm{mod}(\Gamma)\leq b(n)\textrm{mod}(\Gamma_n)$$ where $$b(n)\to 0$$ as $$n \to \infty$$.

I suppose an approach is to try to come up with something that approximates an extremal metric but I can't figure this out.

Since my question is vague, I will give an example. If $$f_n$$ is the map defined by $$f(x,y)=x+ n^{-1}y$$ then $$\textrm{mod}(\Gamma_n)=nl/h= n\cdot \textrm{mod}(\Gamma)$$. Of course this is special because the $$f_n$$ are affine.

There is always the inequality $$K(f_n)^{-1}\textrm{mod}(\Gamma_n) \leq \textrm{mod}(\Gamma) \leq K(f_n)\textrm{mod}(\Gamma_n)$$ but this is not of use because both terms in the product on the right blow up as $$n$$ grows.