To test Conjecture 1 in these cases, we need only use a MapleV.5 script, which you can view. This script generates an instance, computes a degree reverse lexicographic Gröbner basis, and then uses the FGLM conversion algorithm to compute the eliminant, and then finally checks to see if all the roots are real. If not, it prints an error message. Some procedures to do the FGLM conversion are contained in another Maple script, MakeLex.
We use local coordinates which reduce the number of parameters by 2. While this complicates the first degree reverse lexicographic Gröbner basis (Strange, that fewer variables slow the computation!), it greatly expediates the computation of the eliminant.