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No, the conversion of an entity into a VLA object does not survive closing the drawing and getting back in. Even the entity name of an object can change from one drawing session to another.
The way you would do that is by saving the entity's Handle, which does not change, no matter what. It's in the (assoc 5) entry in entity data:
(setq EntityHandle (cdr (assoc 5 (entget (car (entsel "\nSelect object to get its Handle: "))))))
and then you can get the entity name back for whatever purpose with the (handent) function:
(handent EntityHandle)
You can save such things externally to the drawing, with a few different approaches, such as by way of environment variables [look into the (setenv) and (getenv) AutoLisp functions], in which case you should give the names of those an element relating them to the specific drawing, because the same Handle could exist in other drawings, identifying some different entity.
Another way would be to write them out to a plain text file or a .csv spreadsheet file, using the (write-line) function to put them there and (read-line) to get them out. Again, name such a file with something specific to the drawing you're in.
You could put Handles into Text or Mtext parked off to the side somewhere, which would keep everything within the drawing file itself, not requiring anything external as the above options do [which means, for example, you could send a drawing to someone else, and it would have all that information].
I'm sure there are other ways....
Kent Cooper, AIA