FYI: The "Mormon Church" as you describe it is actually a team at BYU that specializes in genetic identification of family relationships.
Which is funded by the Mormons as it's Brigham Young University
One of their geneticists is one of the top in the country and to date, they have identified a number of unknown mummies who have been floating around various museums.
And he's also a high ranking member of the Church of Latter Day Saints
They have also built a genetic family tree. There is all sorts of things one can learn about ancient Egypt this way. (For example, just because historical records say that two people are related, it doesn't mean they are genetically.)
LDS also has strategic partnerships with Ancestry.com and FamilySearch Do I need to connect the dots for you here too?
The BYU team has no interest in it from a religious perspective. King Tut wouldn't provide any additional "religious" information than the other 30+ mummies they have already worked with. They are very interested in it from a scientific perspective which is another way of saying that they are curious as all scientists should be.
I'm not denying their scientific acumen. It's obvious they are very adept at what they are doing. But the reason this program exists in BYU is to find proof that the native americans were really descended from a lost tribe of Jewish settlers who came to America. Even if the head of this program, who is a high ranking Mormon, isn't interested in this; the LDS would use this as a reason to posthumously baptize Tutankhamun.