Theoretically yes, but at the time the gift was made, these kinds of items were treated as personal gifts and therefore the property of the recipient, especially since it was a gift from a family member (a wedding gift to Diana from QEQM) versus a gift from a foreign ruler / foreign country.
I don't recall the sapphire having any known provenance prior to it being gifted to Diana. If Diana's engagement ring was treated a Diana's personal property which was then inherited by her sons,* I doubt if the necklace would be treated differently.
I do suspect that in this day and age that these kinds of jewels are lifetime loans to be returned to the crown upon the death of the recipient, so for example, Kate's sapphire engagement ring is not hers to dispose of how she pleases and will get returned to the vaults upon her death.
* William or Harry or both of them as part of a pool of items for them to decide how to divvy up.