The sublinear conjugacy-class conjecture for primitive wreath-product subgroups

Let $A$ be an almost simple primitive group on $m$ points appearing in the paper's table of final exceptions, and assume that $k(A)<m$. For each primitive subgroup $G\leqslant A\wr S_r$ acting on $n=m^r$ points, write $k(G)$ for the number of conjugacy classes of $G$. Sublinear conjugacy-class conjecture. For every such $G$,

$$k(G)=o(m^r)\qquad\text{as }r\rightarrow\infty.$$

The conjecture seeks an improvement over the general bound $k(G)<n^{1.31}$ in the cited theorem. Earlier estimates give the desired conclusion in some cases, but not all, so the assertion is presented as an open conjecture.