Steps to Reproduce:

1. Rename chest with anvil.
2. Place Named chest.
3. Place another chest (Unnamed).
4. Look in the unnamed chest.
5. Have the named chest be cloned into the same space as the unnamed chest whilst  looking in the unnamed.

(They also don’t keep names when using pick block)

Observed Results:
Items in chest change but name (top left) does not.

At the end of the clip where I close and reopen the chest the name is different after it's reopened and is not changed after being cloned whilst opened. 
Expected Results:
Both items and name change whilst chest is opened.