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

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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...