  LLT polynomials are $q$-analogues of product of Schur functions that are
known to be Schur-positive by Grojnowski and Haiman. However, there is no known
combinatorial formula for the coefficients in the Schur expansion. Finding such
a formula also provides Schur positivity of Macdonald polynomials. On the other
hand, Haiman and Hugland conjectured that LLT polynomials for skew partitions
lying on $k$ adjacent diagonals are $k$-Schur positive, which is much stronger
than Schur positivity. In this paper, we prove the conjecture for $k=2$ by
analyzing unicellular LLT polynomials. We first present a linearity theorem for
unicellular LLT polynomials for $k=2$. By analyzing linear relations between
LLT polynomials with known results on LLT polynomials for rectangles, we
provide the $2$-Schur positivity of the unicellular LLT polynomials as well as
LLT polynomials appearing in Haiman-Hugland conjecture for $k=2$.
