  Our work highlights the crucial role played by the equation of state (EoS)
parameter $w$ within the context of single field inflation with Multiple Sharp
Transitions (MSTs) to untangle the current state of the PBH overproduction
issue. We examine the situation for a broad interval of EoS parameter that
remains most favourable to explain the recent data released by the pulsar
timing array (PTA) collaboration. Our analysis yields the interval, $0.2 \leq w
\leq 1/3$, to be the most acceptable window from the SIGW interpretation of the
PTA signal and where sizeable PBHs abundance, $f_{\rm PBH} \in (10^{-3},1)$, is
observed. We also obtain $w=1/3$, radiation-dominated era, to be the best
scenario to explain the early stages of the Universe and address the
overproduction problem. Within the range of $1 \leq c_{s} \leq 1.17$, we
construct a regularized-renormalized-resummed scalar power spectrum whose
amplitude obeys the perturbativity criterion while being substantial enough to
generate EoS dependent scalar induced gravitational waves ($w$-SIGWs)
consistent with NANOGrav-15 data. Working for both $c_{s} = 1\;{\rm
and}\;1.17$, we find the $c_{s}=1.17$ case more favourable for generating large
mass PBHs, $M_{\rm PBH}\sim {\cal O}(10^{-6}-10^{-3})M_{\odot}$, as potential
dark matter candidates with substantial abundance after constraints coming from
microlensing experiments.
