On a Speculated Relation Between Chv\'atal-Sankoff Constants of Several Sequences

It is well known that, when normalized by n, the expected length of a longest common subsequence of d sequences of length n over an alphabet of size sigma converges to a constant gamma_{sigma,d}. We disprove a speculation by Steele regarding a possible relation between gamma_{2,d} and gamma_{2,2}. In order to do that we also obtain new lower bounds for gamma_{sigma,d}, when both sigma and d are small integers.