Your post shows values of d=2, 3.09, 3.29 and 3.31 with different latices. There must be some theoretical maximum d possible, right? Some upper bound that "d cannot be greater than X"? What do we know about the bounds of that maximum value?
These produce the largest d for N=4,8,6,12 and 20 resp. Thus, it is presumed by almost everyone that d=3.64 is the global upper bound.
Unfortunately, I believe it is still an open problem to to prove this or to describe a general upper bound for specific N not equal to any of these five values.