Central-limit conjecture for the logarithm of the Ulam–Kac adder

Let $(X_n)_{n\geq 0}$ be the Ulam–Kac adder sequence, and let convergence in law be denoted by $\to$. Central-limit conjecture. There are positive constants $\mu$ and $\sigma$ such that

$$\frac{\log X_n-\mu n^{1/2}}{\sigma n^{1/4}}\to\mathcal{N}(\mu,\sigma^2)\text{ in law}.$$

The supplied context identifies this as an open problem in computer science, while the conjectured normalization proposes Gaussian fluctuations for $\log X_n$ around a mean of order $n^{1/2}$ with scale $n^{1/4}$.