While has many other meanings besides the whilst / in the midst of one, which is not true of whilst. For example the noun meaning of while, i.e. a period of time (e.g. abide the while) and the verb ('to while away my time'), etc.
I've often heard this usage...
'Whilst (while) I go to the shops to look around, she goes on for one of her spending sprees'.
It doesn't seem to take on the conjunctive role here, I think. Maybe it's just wrong. I'm eager to hear from the veterans.