Deducing the boss's birthday from month and day hints

A team knows their manager's birthday is one of these ten dates:

• Day 1: September 1 or December 1
• Day 2: December 2
• Day 4: March 4 or June 4
• Day 5: March 5 or September 5
• Day 7: June 7
• Day 8: March 8 or December 8

The manager privately tells one colleague, Bob, only the month, and tells another colleague, Charlie, only the day. Bob then says: "I don't know the date, and I'm certain Charlie doesn't either." Charlie answers: "I didn't know before, but now I do." Bob then says: "Now I know too." On which date is the birthday? Give it as a four-digit number mmdd (with leading zeros).

0901 (September 1). Days $2$ and $7$ occur only once, so Bob can only be sure Charlie is in the dark if his month is March or September. Charlie then knows the date, so his day is unique within those months, ruling out the $5$s and leaving March 4, March 8 and September 1. Bob can now pin it down only if his month has a single option left, which is September.