Didn’t Jesus say in Luke 7:28, that no one born of women, was greater than John the Baptist? I would suspect John was greater than Mary, since were both born of women. Kind of blows their theory out of the water. I don’t expect anyone to pay attention, but hey, we try.
Now there you go reading the text again!
You obviously interpreted that wrong...Check with your local priest, or Imam...
This is made clear by the context. The reference to John’s greatness occurs in a discussion of his role as a prophet (11:7-15). And the declaration of his greatness is occasioned by Jesus quoting the prophecy of the coming prophet (v. 10) who would herald the Messiah (Mal. 3:1). Jesus follows up the remark by pointing to John as the terminus of the Law and the prophets (v. 13) and as the fulfillment of the prophecy that Elijah (or one like Elijah) would come (v. 14).
This explains the sense in which John is greater than all others: He is the greatest of the pre-Christian prophets in the sense that he gets to be the herald of the Messiah himself. Indeed, some manuscripts of Luke’s Gospel give the parallel to this saying as, “I tell you, among those born of women no prophet is greater than John the Baptist” (Luke 7:28, var.). Needless to say, the class of Old Testament prophets is a set to which Mary does not belong.
Further, Jesus’ universe of discourse at this point is limited by another factor: time. “The kingdom of heaven” refers to the Christian age, to the Church that Jesus will found. John thus may be the most blessed of the line of Old Covenant prophets because he is the Messiah’s herald, but even the humblest Christian will be more blessed than John is. And of course Mary-unlike John-lived into the Christian age and became part of the Church.
https://www.catholic.com/magazine/print-edition/the-universe-of-discourse