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A couple of points:
1) LRR is not an absolute term. It means "better fuel economy compared to other tires with the same traction and wear characteristics".
So it is quite possible to get worse fuel economy with a tire labeled LRR than one not so labeled.
2) For tires, there is a technological triangle involving treadwear, traction and rolling resistance. In order to get good values in one area, one or both of the others has to be sacrificed.
3) OEM tires almost always have really good fuel economy, but they do that by sacrificing traction and/or treadwear. That's why OE tires are considered poor quality - the goal was good F/E, not good wear.
The OE tire mention by the OP (Toyo Nanoenergy A29) has a UTQG rating of 300 A B. The treadwear rating is really low.
4) As tires wear, the RR decreases. So getting new tires should always result in a loss of F/E - all other things being equal.
5) There is no break in for RR in tires. The first few thousand miles have fairly rapid wear, so the F/E improves quickly as the tread is worn away, but RR continues to improve until it reaches its best numbers just before being worn out.
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