  We conjecture an explicit positive combinatorial formula for the expansion of
unicellular LLT polynomials in the elementary symmetric basis. This is an
analogue of the Shareshian-Wachs conjecture and previously studied by Panova
and the author in 2018. We show that the conjecture for unicellular LLT
polynomials implies a similar formula for vertical-strip LLT polynomials.
  We prove positivity in the elementary basis in for the class of graphs called
`melting lollipops' previously considered by Huh, Nam and Yoo. This is done by
proving a curious relationship between a generalization of charge and
orientations of unit-interval graphs.
  We also provide short bijective proofs of Lee's three-term recurrences for
unicellular LLT polynomials and we show that these recurrences are enough to
generate all unicellular LLT polynomials associated with abelian area
sequences.
