Will the gum around the top of the crown on front right tooth (#8) eventually settle down around the crown over time? (photos)

Hello thank you for posting your pictures and questions, I'm Dr. Pedro Ramirez specialized dentist from Colombia
Every time that you a start treatment in your mouth you have to keep in mind to aspects of it, your dental health and your dental appearance, In this case, the crown is pushing the gum and this could make your gum looks swollen, there are a lot of patients who have some problems with their gums and you have to verify if you are one of them, you can resolve this problem with a correct oral cleaning however i suggest to set an appointment with the dentist and verify if you need a gum Contour Surgery.

because as you say the crown is pushing the gum.It is good to know if they made a contour of the gums ?, If so or not, fraronron the crown to where it is now, because the preparation of the tooth reaches that level and that is why it looks like it is pushing the gum.
good luck Dr. Ruben Mateo