  We investigate unification prospects of two heavy scalar triplet extension of
the standard model where, in the absence of any right-handed neutrino (RHN),
type-II seesaw accounts for current oscillation data with hierarchical neutrino
masses consistent with cosmological bounds and the lighter triplet decay
explains baryon asymmetry of the Universe via leptogenesis. We note that the
absence of RHNs in the fundamental fermion representations of SU(5) delineates
its outstanding position compared to SO(10) (or $E_6$). In addition, SU(5)
needs smaller scalar representations ${15}_{H1}\oplus {15}_{H2}$ compared to
much larger representations ${126}_{H1}\oplus {126}_{H2} \subset $ SO(10) (or
${351}^{\prime}_{H1}\oplus {351}^{\prime}_{H2} \subset E_6$). We show how
precision gauge coupling unification is achieved through SU(5) with the
predictions of different sets of two heavy triplet masses which, besides being
compatible with type-II seesaw, are also consistent with unflavoured or $\tau
-$ flavoured leptogenesis. In addition to an intermediate mass colour octet
fermion, completion of precision gauge coupling unification is found to require
essentially the presence of the well known weak triplet fermion $\Sigma
(3,0,1)$ in its mass range $M_{\Sigma}\simeq {\cal O}(500-3000)$ GeV out of
which the dominant dark matter (DM) resonance mass $M_{\Sigma}\ge 2.4$ TeV is
known to account for the observed cosmological relic density. The deficiency in
relic density for lighter $\Sigma$ solutions is compensated by an additional
scalar singlet DM. A GUT ansatz is noted to ensure vacuum stability of the SM
scalar potential for all types of unification solutions realised in this work.
We discuss proton lifetime estimations for $p\to e^+\pi^0$ compatible with the
present Hyper-Kamiokande bound as a function of an unknown mixing parameter in
the model.
