I'm not sure how in depth one can go here. There is no consensus. There is broad agreement that A) they are actual love letters, rather than imitation love letters for rhetorical practice, and B) that they are from at least roughly the right context (i.e. northern France in the twelfth century). They seem to be most closely connected with the Loire poets, particularly Baldric of Dol and Marbod of Rennes; and there doesn't seem to be any uncontroversial references that would place them beyond the first quarter of the century. But equally, there is no smoking gun. We have other such love letters from the period, most famously in the Tegernseer Briefsammlung, which also incidentally contains some of the few (only?) surviving medieval, lesbian love letters. (Which are amazing, I highly recommend!)^1
So for the majority of scholars it comes down to more wishy-washy stylistic and contextual issues, with proponents mostly stressing the close contextual similarity and lack of known alternatives that fit the criteria, while the skeptics stress certain discrepancies and plausible gaps in our knowledge of the period.
To be more concrete. In her translation of these and some similar letters like Tegernseer love letters, Barbara Newman lists the state of the field as such:
The major proponents of authenticity are Newman herself, C. Stephen Jaegar and Constance Mews, along with some of older figures like Karl Langosch, Walther Bulst and the first editor, Eward Könsgen. The most strenuous opponent of authenticity is Peter von Moos, although I don't think his argument is taken especially seriously by most – except apparently Peter Godman in some comments to the Times Literary Supplement. Most of the other voices oppose the attribution on the grounds of varying degrees of skepticism. These include Jan Ziolkowski, Giovanni Orlandi, Francesco Stella, Giles Constable, Peter Dronke, as well as earlier figures like Bernhard Bischoff and André Vernet. (Newman, p. xiii)
For my money, the stylistic disanalogies that Ziolkowski^2 points to are more convincing than the contextual similarities that Newman stresses. But lacking some further evidence or some new way of interpreting the EDA, most seem to agree that we can't get further than plausible attribution or plausible skepticism.
1: They're freely accessible in the MGH if you read Latin (Liebesbriefe, n.6-8), otherwise they're translated in Barbara Newman, Making Love in the Twelfth Century (2016) (n. 5-7 in Newman).
2: "Lost and Not Yet Found: Heloise, Abelard, and the Epistolae duorum amantium", Journal of Medieval Latin 14 (2004): 171-202.