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If Chris hopes to survive in his 8 a.m. English class, he needs to discover
caffeine even when he has a test,
his head hits the desk by 8:05, and snores soon escape his mouth.
The comma between test and he is completely correct. Even when he has a test is a subordinate clause, so it cannot stand alone. His head hits the desk by 8:05 is a main clause.
When a subordinate clause introduces a main clause, you need to connect them with a comma.