The Skolem conjecture for simple rational linear recurrence binary sequences

Let $\boldsymbol{u}=(u_n)_{n\in\mathbb{Z}}$ be a simple rational LRBS, meaning a simple rational linear recurrence binary sequence, taking values in $\mathbb{Z}[1/b]$ for some integer $b$. An integer zero is an index $n\in\mathbb{Z}$ such that $u_n=0$, and $\gcd(b,m)=1$ means that $b$ and $m$ are coprime. The Skolem conjecture. The sequence $\boldsymbol{u}$ has no integer zero if and only if there exists an integer $m\geq 2$ with $\gcd(b,m)=1$ such that

$$u_n\not\equiv 0\pmod m$$

for every $n\in\mathbb{Z}$. This is also called the Exponential Local-Global Principle and gives a modular certificate for the absence of integer zeros. The supplied source does not provide evidence resolving the conjecture.