Michael Elsby says no. In his view, if downward nominal wage stickiness is a potential problem, the relevant class of firms will simply start workers off at a lower nominal wage, raising it over time as need be. The result will be "wage compression."
Elsby also claims that the Phillips curve trade-off between inflation and unemployment is not stronger at low levels of price inflation. That suggests that nominal wage rigidity doesn’t much matter at the macro level. If it did, proximity to that zero nominal cut point ought to boost the benefits of inflation, but it doesn’t seem to.
I am surprised by this argument, but I don’t (yet?) see reason to reject it.