Are folds that maximize the number of H-H contacts the best folds for a prototein? Not necessarily.
A high H-H score enhances a molecule's stability, which is certainly a useful property in a biopolymer, but there are other factors to consider as well. Stability implies that once a molecule is folded, it will probably stay folded. It's also important, however, that all molecules with the same sequence fold up to yield the same structure. The way to achieve such uniformity is to select sequences that have a unique best folding, even if that folding does not have the highest possible H-H score.
A molecule with many equally good foldings is said to have a degenerate ground state. The all-P sequence is an obvious example: Every folding has an H-H score of zero. The all-H sequence is also degenerate. Obviously, any sequences with unique preferred foldings must be found between these extremes, but the existence of such sequences cannot be taken for granted. You can search for them by sorting all the foldings of a sequence into bins according to their H-H score; if the highest-scoring bin has a single occupant, that sequence has a unique best folding.
On the square lattice, uniquely folding sequences do exist for all but one of the chain lengths I was able to test. (The exception is length 5.) The longest chains I examined have 14 beads. Among the 16,384 sequences of this length, 955 have a unique folding. Within this subset, 96 foldings have seven H-H contacts, which is the maximum observed in 14-bead prototeins. The sequences in this elite subset, combining uniqueness with high stability, might be considered among the most lifelike prototeins.
Low degeneracy and numerous contacts are not the only criteria for judging a prototein fold. Martin Karplus and Eugene Shakhnovich work with three-dimensional lattice models and employ a more realistic energy spectrum than the simple contact counting of the H-P scheme. Their findings highlight the importance of having a large energy gap between the best folding and the next-best one. They have also looked into the kinetics of folding, asking not just which configuration is stablest but also how long it takes a randomly wriggling molecule to find that conformation. Among 200 candidate sequences, 30 repeatedly discovered the state of minimum energy after no more than 50 million small random rearrangements.