inflation to some small value [$V \ll K$]{} after inflation. The transition itself would actually trigger the end of inflation, leading to a post-inflationary EoS [${\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu} \simeq 1 - \mathcal{O}(V/K)$]{}. In general we expect that the EoS can approach unity from below, but never achieve ${\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu} = 1$ exactly, as this would require an exactly flat direction with $V = 0$. A natural scenario where inflation is followed by a KD phase is that of Quintessential-Inflation [@Peebles:1998qn], where the inflaton potential $V(\phi)$ is engineered so that the necessary transition occurs at the end of inflation, and the potential is also adjusted to describe the observed dark energy as a quintessence field, see e.g. [@Peloso:1999dm; @Huey:2001ae; @Majumdar:2001mm; @Dimopoulos:2001ix; @Wetterich:2013jsa; @Wetterich:2014gaa; @Hossain:2014xha; @Rubio:2017gty] for different proposals. As mentioned before, a stiff period can be also engineered through the oscillations of the inflaton with potential $V \propto \phi^{2n}$, $n \geq 3$. In this case we note however that the stiff period cannot be sustained for very long, as self-resonant effects lead eventually to a fragmentation of the coherent oscillating inflaton condensate [@Lozanov:2017hjm].

Considering the presence of an SD period before the onset of RD is actually not only theoretically 