Conjecture on conjugacy-class intersections in closed surface groups

Let $\Gamma_g$ be the Cayley graph of a closed surface group with respect to a standard generating set. For a conjugacy class $\mathfrak{c}$, let $\lVert\mathfrak{c}\rVert$ denote the minimum word length of a representative of $\mathfrak{c}$, and let $B_n$ be the radius-$n$ ball in $\Gamma_g$. Conjugacy-class intersection conjecture. There is a polynomial $p$ such that

$$\max_{\mathfrak{c}} \#(\mathfrak{c}\cap B_n)\leq p(\lVert\mathfrak{c}\rVert)\cdot \#\left(B_{(n-\lVert\mathfrak{c}\rVert)/2}\right).$$

For a fixed conjugacy class in a hyperbolic group, the exponential growth rate is half that of the ambient group, but obtaining a uniform bound over all conjugacy classes is more delicate because the relevant centralizer estimates depend on the word length of the class. The conjecture would, together with the paper's lifting arguments, imply an analogue of the simple lifting degree result for closed surfaces.