Fix $m.$ For each $n$ consider the the question: among  all the conjugacy classes for permutations in $S_n$ with exactly $m+1$ cycles, which has the largest size?

I will write $\prod_{i \geq 1}i^{a_i}$ to describe a partition with $a_i$ cycles of length $i.$

Below is some data. It is the result of rather unsophisticated exhaustive calculation. It strongly suggests  the following unsurprising result:

> There is a $k=k(m)$  integers $a_1 \geq a_2 \geq \cdots \geq a_k=1$ and $t=t(m)=\sum_1^ka_kk$ such that, for $n$ large enough relative to $m,$ the unique maximum occurs for type $1^{a_1}2^{a_2}\cdots k^{a_k}(n-t)$

The particular counts do not seem to support the conjecture from the question. 

- For $m=1$ the maximum occurs for $1\ (n-1)$
- For $m=2$ and $n \gt 5$ the maximum occurs for $1\ 2\ (n-3)$
- For $m=3$ and $n \gt 10$ the maximum occurs for $1^2\ 2\ (n-4)$
- For $m=4$ and $n \gt 10$ the maximum occurs for $1^2\ 2\ \ 3 (n-7)$
- For $m=5$ and $n \gt 20$ the maximum occurs for $1^3\ 2\ 3\ (n-8)$
- For $m=6$ and $n \gt 17$ the maximum occurs for $1^3\ 2\ 3\ 4\ (n-12)$ **except that** 
 -  for $n=22$ there is a tie between  $1^3\ 2\ 3\ 4\ (10)$ and $1^2\ 2\ 3\ 4\ 5\ 6$
- For $m=7$ and $n \gt 29$ the maximum occurs for $1^3\ 2^2\ 3\ 4\ (n-14)$

This might allow a more refined conjecture. Certainly data for larger $m$ would be suggestive. That would justify taking the time to do design a more intelligent search. 

Here are more detailed results on $m=7$ to show exactly what I know for sure:

For $30 \leq n \leq 60$ the maximum occurs only for $1^3\ 2^2\ 3\ 4\ (n-14).$ Presumably the same is true for all $30 \leq n.$ It is also true for $n=19,20,21,22. $

the other cases are:

 - $1^8$
 - $1^72$
 - $1^62^2$
 - $1^62\,3$
 - $1^52^23$
 - $1^42^33$
 - $1^5\,2\,3\,4$
 - $1^42^23\,4$
 - $1^42^23\,5$
 - For $n=17$ a tie between $1^42^23\,6$ and $1^32^23^24$
 - $1^4\,2\,3\,4\,5$ for $n=18.$
 - $1^32\,3\,4\,5\,(n-17)$ for $23 \leq n \leq 28$
 - for $n=29$ a tie between $1^3\,2\,3\,4\,5\,12$ and $1^32^23\,4\,15$