Can constant-ambiguity CFGs strictly generalize unambiguous CFGs?

Determine whether the class of context-free languages that admit a context-free grammar in which every accepted word has exactly the same number of derivation trees for all words (i.e., constant per‑word ambiguity) strictly contains the class of unambiguous context-free languages; equivalently, show whether there exists an ambiguous context-free language for which each accepted word has exactly c ≥ 2 derivations per word.

Background

In discussing their CYK-style dynamic program for probabilistic membership on unambiguous CFGs, the authors note that the method would also work for grammars where each accepted word has the same number of parse trees, because no double counting arises if that number is constant. They then raise the question of whether this class is strictly larger than uCFLs.

The issue concerns the structure of ambiguity in CFGs: if a grammar has constant ambiguity c across all accepted words, is there an example with c>1 that cannot be recognized by any uCFG? Resolving this would clarify whether their algorithmic technique extends beyond uCFLs in a principled way.