@Giovannini:2008tm; @Cui:2017ufi; @Caprini:2018mtu; @Cui:2018rwi], opening up the possibility of detection of this GW background by upcoming GW direct-detection experiments. Furthermore, the possibility of a stiff epoch is also well motivated on the theory side, by various early Universe scenarios for which the implementation of a post-inflationary stiff period is crucial. For instance, in [*Quintessential inflation*]{} [@Peebles:1998qn; @Peloso:1999dm; @Huey:2001ae; @Majumdar:2001mm; @Dimopoulos:2001ix; @Wetterich:2013jsa; @Wetterich:2014gaa; @Hossain:2014xha; @Rubio:2017gty] the inflationary epoch is followed by a period where the universe is dominated by the kinetic energy of the inflaton, with the potential adjusted to describe the observed dark energy as a quintessence field. In the original [*gravitational reheating*]{} formulation [@Ford:1986sy; @Spokoiny:1993kt] the Universe is reheated by the decay products of inflationary spectator fields if the Universe undergoes a sufficiently long stiff epoch after inflation. Even though basic implementations of such gravitational reheating scenarios have been shown to be inconsistent with BBN/CMB constraints [@Figueroa:2018twl], unnatural $ad~hoc$ constructions are still viable. Furthermore, variants that can naturally avoid the inconsistency have been also proposed, like the [*Higgs-reheating*]{} scenario [@Figueroa:2016dsc], where the Standard Model (SM) Higgs is a spectator field with a non-minimal coupling to gravity and the Universe is reheated into SM relativistic species (decay products of the 