Just to mark this question as answered: The answer is yes. Some details follow.

The basic idea was generously provided by the anonymous referee of a short note (joint work with Paolo Leonetti) that has been recently accepted for publication in some journal (*). The key ingredient is Theorem 3 from:

J.-H. Evertse, The number of solutions of decomposable form equations, Invent. Math. 122 (1995), No. 3, 559–601,

which yields, for a fixed finite set of primes $\mathcal S$, an effective bound on the number of non-degenerate solutions of an $\mathcal S$-unit equation in $k$ variables (over the additive group of the rationals).

(*) I'm still trying to understand how to avoid self-promotion in situations like this...

Just in order to mark this question as answered: The answer is yes. Some details follow.

The basic idea (for some more general conclusion) was generously provided by the anonymous referee of a short note (joint work with Paolo Leonetti) that has been only recently accepted for publication in JNT (*). The key ingredient is Theorem 3 from:

J.-H. Evertse, The number of solutions of decomposable form equations, Invent. Math. 122 (1995), No. 3, 559–601,

which yields, for a fixed finite set of primes $\mathcal S$, an effective bound on the number of non-degenerate solutions of an $\mathcal S$-unit equation in $k$ variables (over the additive group of the rationals).

More precisely, Evertse's theorem implies the existence of a base $\theta \in \mathbf R^+$ such that $\omega(s_n) \gg \text{slog}_\theta(n)$ for infinitely many $n$, where $\text{slog}_\theta$ is a kind of inverse of tetration (the same can be also proved by using only elementary means, but this question was about the Subspace Theorem and its descendants).

(*) I hope this doesn't sound as self-promotion. If it does and you have any suggestion on how to avoid it in cases like this, then I'd appreciate to know.