Full Idea
It is the core essential features of the object that will be independent of how things turn out, and they will be independent in the sense of holding regardless of circumstances, not whatever the circumstances.
Gist of Idea
Essential features of an object have no relation to how things actually are
Source
Kit Fine (Necessity and Non-Existence [2005], 09)
Book Reference
Fine,Kit: 'Modality and Tense' [OUP 2005], p.349
A Reaction
The distinction at the end seems to be that 'regardless' pays no attention to circumstances, whereas 'whatever' pays attention to all circumstances. In other words, essence has no relationship to how things are. Plausible. Nice to see 'core'.