Homotopy equivalence preserving all geometric intersection numbers This question again might be silly, like the last post(deleted). Let me know I will delete it.

Problem: Let $\Sigma$ be a surface without boundary and $f:\Sigma\to \Sigma$ be
a proper homotopy equivalence. Suppose for all closed curves
$\alpha,\beta:\Bbb S^1\to\Sigma$ we have
$i\big([f\circ\alpha],[f\circ\beta]\big)=i\big([\alpha],[\beta]\big)$,
i.e. $f$ preserves all geometric intersection numbers. Is it true that
$f$ is properly homotopic to a homeomorphism?

Note that if $\Sigma$ is a closed surface, then any homotopy equivalence is homotopic to a homeomorphism. So the problem is clear without the extra assumption: "geometric intersection number is preserved."
Also, note that any homeomorphism of $\Sigma$ preserves the geometric intersection number.
I am not sure about the term "proper," i.e., the problem might be well-posed if one replaces proper homotopy equivalence with ordinary homotopy equivalence and proper homotopy with ordinary homotopy. I used the term proper keeping in mind the open surface.

Even in the closed surface case, is it possible to prove the above problem a priori not assuming "every homotopy equivalence is homotopic to a homeomorphism"?

Any help will be appreciated. Thanks in advance.
 A: The answer is "no". $\newcommand{\CC}{\mathbb{C}}$
Consider the map $f \colon \CC \to \CC$ given by $f(z) = z^2$.  All geometric intersection numbers in $\CC$ are zero, so the extra assumption holds automatically.  Note that $f$ is proper, and is a homotopy equivalence.
However, there is no proper homotopy of $f$ to a homeomorphism.
EDIT (after the comments below and above):
As I said, above I am assuming that $f$ is proper, and is a homotopy equivalence. But the original poster has clarified that what is wanted is that $f$ is proper, is a homotopy equivalence, has a homotopy inverse $g$ (which is proper), and the compositions $f \circ g$ and $g \circ f$ are properly homotopic to the identity.
In this case, if $\Sigma$ has finite topological type, then $f$ is indeed properly homotopic to a homeomorphism.  This is part of the Dehn-Nielsen-Baer theorem - see Theorem 8.8 of the "Primer on mapping class groups".  The assumption on geometric intersection numbers is not needed.
[When $\Sigma$ has infinite topological type, I don't know the answer.  But I guess that $f$ is again properly homotopic to a homeomorphism, perhaps via some exhaustion argument?]
