Stojnić et al. (2013, 2017) argue that the reference of demonstratives is fixed without any contribution from the extra-linguistic context. On their `prominence/coherence' theory, the reference of a demonstrative expression depends only on its context-independent linguistic meaning. Here, we argue that Stojnić et al.’s striking claims can be maintained in only the thinnest technical sense. Instead of eliminating appeals to the extra-linguistic context, we show how the prominence/coherence theory merely suppresses them. Then we ask why one might be tempted to try and offer such a view. Since we are rather sympathetic to the motivations we find, we close by sketching a more plausible alternative.