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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.

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