On the basis of a variable neighbourhood search with the AutoGraphiX software, it is conjectured that for even numbers of atoms the fully conjugated acyclic pi system of maximum HOMO-LUMO gap is a 'comb' in which each vertex of a backbone carries a single pendant edge. Chemically, this represents CnH3n/2+2, an alpha, omega -diene with methylene groups attached at all intermediate positions. As n increases without limit, the conjectured asymptotic dimensionless gap is 2 root2 - 2. Odd-numbered graphs that would have maximum gap as singly charged pi systems are reported for up to 41 vertices; they are related to comb graphs.