The simplest models of inflation are so-called canonical ("normal" kinetic energy term), single field (a single field, the inflaton, is responsible for both driving the inflationary expansion and generating the initial density perturbations), slow roll (smooth, gradually decreasing potential energy function) models. These generically predict near-spatial flatness and a nearly scale invariant spectrum of Gaussian, adiabatic density perturbations. Primordial gravitational waves may or may not be generated at observable levels by these models. If any of these predictions are disconfirmed, then canonical, single field, slow roll inflation is false.
Inflation is a broad and phenomenologically rich paradigm, and there's much more to it than these simplest of models. If the universe turns out not to be near-critical, then there's a model for this: so-called open inflation which was studied by Linde and others in the 90's before the universe was seen to be accelerating. If the perturbations are non-Gaussian, then depending on the type, either the kinetic term is noncanonical, there is transient non-slow roll behavior, or there are multiple fields contributing to the generation of perturbations (the curvaton, for example). If the spectrum is not nearly-scale invariant, then we either have non-slow roll behavior across some range of scales, or there are multiple fields driving inflation such that there are rather abrupt changes in "direction" in field space. Another possibility is hybrid inflation which can generate a very blue spectrum ($n > 1.5$), but these models require an additional auxiliary field to end inflation, and so are not single field models. If the perturbations are non-adiabatic, then there's likely an isocurvature component, for instance, via the curvaton mechanism.
The point being that there are enough knobs to turn across the inflationary model space to accommodate a wide range of predictions. But, the "generic" predictions: flatness, near scale invariance, Gaussian, adiabatic perturbations are the hallmark of the most basic and best studied family of models. It is doubtless that this is the inflation that Guth, Linde and others are referring to in their statement, if not explicitly stated.