Fibonomial Lucas-type divisibility conjecture

Let $F_n$ denote the Fibonacci numbers, and define the Fibonomial coefficient by

$${n \choose k}_F=\frac{F_nF_{n-1}\cdots F_{n-k+1}}{F_kF_{k-1}\cdots F_1}.$$

For a prime $p$, let $p^*$ be the least positive integer such that $p\mid F_{p^*}$, and let

$$\mathcal{F}_p=(1,p^*,p^*p,p^*p^2,\ldots).$$

Write $(n)_{\mathcal{F}_p}=(n_i)_{i\geq 0}$ and $(k)_{\mathcal{F}_p}=(k_i)_{i\geq 0}$.

Fibonomial Lucas-type divisibility conjecture. If $p^*\geq p$, then

$$p\mid {n\choose k}_F\quad\Longleftrightarrow\quad p\mid {n_0\choose k_0}_F{n_1\choose k_1}_F{n_2\choose k_2}_F\cdots.$$

This conjecture proposes that, when the Fibonacci entry point $p^*$ is at least $p$, divisibility of a Fibonomial coefficient by $p$ is determined by the corresponding digitwise product in the base $\mathcal{F}_p$. The preceding discussion notes that the analogous relation holds for $p=2$, while for $p^*<p$ it can fail even to preserve divisibility; the status for the stated range is left open here.