Is $w \neq -1$ evidence for a dynamical dark energy equation of state ?

Current constraints on the dark energy equation of state parameter, $w$, are expected to be improved by more than one order of magnitude in the next decade. If $|w-1| \gsim 0.01$ around the present time, but the dark energy dynamics is sufficiently slow, it is possible that future constraints will rule out a cosmological constant while being consistent with a time-independent equation of state parameter. In this paper, we show that although models with such behavior can be constructed, they do require significant fine-tuning. Therefore, if the observed acceleration of the Universe is induced by a dark energy component, then finding $w \neq -1$ would, on its own, constitute very strong evidence for a dynamical dark energy equation of state.