Fragments / Shards could potentially be added for every milestone achieved in GE also, similar to how each guild member recieves fragments of a selection kit upon the completion of a GBG round. GElvl1 could offer 1-1-1-2, GElvl2 could offer 2-2-2-4, Gelvl3 could offer 4-4-4-8, and GElvl4 could offer 8-8-8-16, which could provide the guarantee of fragments that the GBG does for the SoH for example.
I see the logic, but those may be slightly too many 'guaranteed' shards for a single GE (if there are other sources for fragments as well). And making them part of the 'chance reward' from gold reward boxes for 'grand challenges' isn't going to go down well as it would likely reduce the chance of winning diamonds.
Might be easier to make it even more like GBG by creating 'one weekly amount' for final GE placement. There's the same number of guilds competing in GE and GBG, if memory serves. So could just take 50% (for it being 6 days rather than 11) of the shards awarded in Gold league (as a median). Something like 50 shards for 1st, 45 for 2nd, 40 for 3rd etc. awarded on Monday morning when GE ends. That would give individual players a vested personal interest in the final placement of their guilds as well.
In addition, fragments could then be added as a 'chance' from green relics and maybe one in every 8 challenge chests on the first 3 levels, for example the 7th and 15th, giving the chance of 1, 2, or 3 fragments OR 1, 2, or 3 units.