  We study a class of leptogenesis scenarios with decay or scattering being the
source of lepton asymmetry, which can not only give rise to the observed baryon
asymmetry in the universe but also can leave behind a large remnant neutrino
asymmetry. Such large neutrino asymmetry can not only be probed at future
cosmic microwave background (CMB) experiments but is also motivating due to its
possible role in solving the recently reported anomalies in $^4{\rm He}$
measurements. Additionally, such large neutrino asymmetry also offers the
possibility of cogenesis if dark matter is in the form of a sterile neutrino
resonantly produced in the early universe via Shi-Fuller mechanism. Considering
$1 \rightarrow 2, 1 \rightarrow 3$ as well as $2 \rightarrow 2$ processes to be
responsible for generating the asymmetries, we show that only TeV scale
leptogenesis preferably of $1 \rightarrow N \, (N \geq 3)$ type can generate
the required lepton asymmetry around sphaleron temperature while also
generating a large neutrino asymmetry $\sim \mathcal{O}(10^{-2})$ by the epoch
of the big bang nucleosynthesis. While such low scale leptogenesis can have
tantalising detection prospects at laboratory experiments, the indication of a
large neutrino asymmetry provides a complementary indirect signature.
