The asymptotic Cobham conjecture for regular sequences

Let $k,\ell\in\mathbb{N}$ be multiplicatively independent, meaning that no positive powers of $k$ and $\ell$ are equal. Let $\mathbf a=(a_n)_{n\in\mathbb{N}_0}$ be an integer sequence that is both asymptotically $k$-regular and asymptotically $\ell$-regular. Asymptotic Cobham conjecture. The sequence $\mathbf a$ satisfies a linear recurrence almost everywhere: there exist $d\in\mathbb{N}$ and $t_1,t_2,\dots,t_d\in\mathbb{Z}$ such that

$$a_{n+d}=t_1a_{n+d-1}+t_2a_{n+d-2}+\dots+t_da_n$$

for almost all $n\in\mathbb{N}_0$. This is proposed as the analogue of Cobham's theorem for asymptotically regular sequences, and the supplied source gives no resolution of the conjecture.