Ben-Naim–Krapivsky moment-growth conjecture for the Ulam–Kac adder

Let $(X_n)_{n\geq 0}$ be the Ulam–Kac adder sequence, and write $f(n)\sim g(n)$ when $\lim_{n\to\infty}f(n)/g(n)=1$. For each positive integer $m$, let $c_m$ denote the constant governing the logarithmic growth of the $m$th moment. Ben-Naim–Krapivsky's moment-growth conjecture. There is an increasing sequence of positive constants $(c_m)_{m\geq 1}$ such that

$$\log E[X_n^m]\sim c_m\sqrt{n}\quad\forall m\geq 1.$$

This conjecture is based on numerical evidence and extends the known first- and second-moment asymptotics for the Ulam–Kac adder; its general validity remains open.