There's some basic literature on the topic here: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1010552 . Slightly improved uniquely mapped reads; though mapping parameters appear to have a stronger impact than choice of reference.

However, for differential expression (BL6/J untreaded vs BL6/J treated) the question isn't so much "how do the quantifications change" but "how do the logFCs change" -- and I don't see published results on this. I should imagine that, by aligning to the strain transcription, some genes might get slightly higher coverage to bump them over the soft-filtering threshold; but few (if any) logFC values should alter based on the reference.

It's probably worth doing twice just to put your mind at ease.