Hinman's generalized period conjecture for Ulam sequences U(1,n)

For integers $1\leq a<b$, let $U(a,b)$ be the Ulam sequence whose later terms are the smallest integers representable as sums of two distinct prior terms in exactly one way. For a positive real number $\lambda$, reduction modulo $\lambda$ is taken using representatives in an interval of length $\lambda$. Hinman's generalized period conjecture. For every integer $n\geq2$, there exist real numbers $\lambda_n$ and $K_n$ such that, for every $\epsilon>0$,

$$U(1,n)\cap[K_n,\infty)\mod\lambda_n\subset\left(\frac{\lambda_n}{3}-\epsilon,\frac{2\lambda_n}{3}+\epsilon\right).$$

Furthermore, for $n\geq4$, one can take $\lambda_n=3n+\lambda'$, where $\lambda'\approx0.417031$. This folklore generalization is supported by numerical evidence; the asymptotic statement is not proved in the source.