Universal Positivity Set

Can one construct an infinite recursively enumerable set $S\subset \mathbb{N}$, for which one can decide: given any linear recurrence sequence $(u_n)$, whether $\exists n\in S$, s.t. $u_n<0$ ? In other words: is there a set of indices for which the positivity problem is decidable? The analogous question where $u_n<0$ is replaced by $u_n=0$ has a positive answer for simple sequences; Luca, Ouaknine and Worrell have constructed such a set explicitly (called the universal Skolem set).