Quadratic growth conjecture for Young graph equivalence classes

Let $Y(g,k)$ denote a Young graph, and let $\operatorname{Classes}(x)$ denote the number of equivalence classes occurring among all $Y(g,k)$ with $g\le x$. Quadratic growth conjecture. The function $\operatorname{Classes}(x)$ grows roughly proportionately to $x^2$. This is an empirical growth claim about the number of Young graph equivalence classes; the source provides no precise asymptotic formulation or resolution.