Generalized Ulam–Kac moment-bound conjecture

Consider the generalized Ulam–Kac adder

$$X_{n+1}=X_n+\gamma X_{U(n)},\qquad X_0=1,$$

where $\gamma$ is the parameter in the source's generalized recurrence. Let $c_m$ denote the corresponding moment-growth constants, and let Theorem~ denote the theorem whose bounds are being generalized. Generalized moment-bound conjecture. Theorem~ holds for this recurrence with the bounds on $c_m$ replaced by

$$[f_1(m,\gamma),f_2(m,\gamma)]$$

for suitable functions $f_1$ and $f_2$. The source notes that first- and second-moment logarithmic growth of order $\sqrt n$ is already known for this generalized adder, while suitable all-moment bounds are conjectural.