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 system of maximum HOMO-LUMO gap is a 'comb' in which each vertex of a backbone carries a single pendant edge. Chemically, this represents C_nH_3n/2+2, an ,-diene with methylene groups attached at all intermediate positions. As n increases without limit, the conjectured asymptotic dimensionless gap is $2sqrt2-2$. Odd-numbered graphs that would have maximum gap as singly charged systems are reported for up to 41 vertices, and are related to the comb graphs.