HQ and NQ stat gap needs to close (Both should allow max stats with materia):

[Katlyna]

I use an HQ main, level 52 tool to craft a level 55 recipe at level 56. I can't get it over 17% Quality (I have tried this)
I use an NQ Main, level 55 tool to craft a level 55 recipe at level 56, I can get it to 100% Quality. (look at the 52 HQ main hand and the 55 NQ - the stats are the same)

As Caimie and Techno have already pointed out your statement is inaccurate. I tested this theory out during the first week of heavensward as I was one of the first insane pushing hard to max all 60 on our server. During the process we were doing wacky stuff like making HQ gear replacements while 2-3 levels under the required recipe with main and offhands which were also under optimal level. Success is purely driven by the stats of your overall gear, not by the iLevel. This has always been true for crafting as far as I've known (fairly sure this was true in beta too). Quality gains specifically are driven by your paper doll control stat listing (including any inner quiet modifiers), the difference between your level and the recipe level, the current step <condition>, and the skill chosen to use. Likewise these can be modified by certain skills to amplify the effects.

---

I will admit when you posed your first comment Ksenia, one assumption I've made in my first post is that V materia won't be available at reasonable prices in patch 3.1. If V materia is available is available and doesn't cost an arm and a leg then NQ two star items could potentially be maxed out using V materia. But I would bet all the nana bears in the world that V materia prices will stay super high during 3.1 for this to be a valid consideration.