On counting centralizer subgroups of symmetric groups

Let $S_{2m}$ be the symmetric group, $h=(1\ 2)(3\ 4)\cdots(2m-1\ 2m)$ and $H=C(h)$. We consider the structure of $gHg^{-1}\cap H$ for any $g\in S_{2m}$. We prove the permutations $g$ which makes $gHg^{-1}\cap H$ have size of polynomial in $m$ have density zero.