grammaticality - Is "Whom should I give this job to?" grammatically correct?

I found a grammar rule about when to choose "who" and "whom".

Rule. Use this he/him method to decide whether who or whom is correct:

he = who

him = whom

So I have a question about correctness of the following question:

Who should I give this job to?

According to this rule we should have used whom instead of who, because I should give this job to him, not I should give this job to he. But there's a man who keeps telling me that

Whom should I give this job to?

is wrong. Please clarify?