Positive order polynomials for width-two and shifted or cylindric skew-shape posets

Let $\Omega(P;t)$ denote the order polynomial of a finite poset $P$. The classes under consideration are posets of width two, cell posets of cylindric skew shapes, and cell posets of shifted skew shapes; these classes are closed under taking lower and upper ideals. Positivity conjecture. The order polynomials of all three classes have nonnegative coefficients: posets of width two, cell posets of cylindric skew shapes, and cell posets of shifted skew shapes. The metatheorem described immediately before this conjecture reduces the problem for these stable classes to nonnegativity of the linear coefficient. The source presents the claims as conjectural.