Hollander's conjecture on regular numeration languages and dominant-root limits

Let $U=(U_n)_{n\geq 0}$ be the recurrence sequence defining the numeration system, let $L$ be its numeration language, and let $P$ be the minimal polynomial of the recurrence relation satisfied by $U$. A sequence is regular when its numeration language has the regularity property considered in the paper.

Hollander's conjecture. If $L$ is regular, there exists $p$ such that the limit

$$\lim_{n\to\infty} \frac{U_{np+i}}{U_{(n-1)p+i}}$$

exists and is independent of $i$. Furthermore, the minimal polynomial $P$ is of the form

$$P(X)=Q(X^p),$$

where $Q$ is the minimal polynomial for a recurrence that gives rise to a regular language.

This conjecture, attributed to Hollander, relates regularity of a numeration language to the asymptotic behavior of the defining recurrence and to the structure of its minimal polynomial. The supplied material does not state whether it has been proved or disproved.