Re. the bolded - I am confident that that is a safe assumption: we consumers never benefit from less-than-complete information from less-than-reliable authorities. Apples-to-apples is obviously impossible by definition unless both stones have reports from the same lab, but an independent appraisal will tell you what you need to know for the purposes of this discussion..
Let me clarify what I meant by apples-to-apples. Several on this board have suggested that cutters often use EGL-I to certify otherwise sub-par diamonds in order to give them some (albeit false) competitiveness with their GIA and AGL counterparts. Based on discussion here, I'm led to believe that on average an EGL-I certified diamond will price higher than had the diamond been GIA certified. If I'm misunderstanding this, please correct me.