Does the apparent radius of the event horizon of a black hole depend on the distance of the observer from the black hole?

Alice, Bob, and Charlie are sitting on a stationary platform a long way from a black hole. Alice tells Bob and Charlie to go jump in the black hole. Bob grabs a clock off the mantelpiece and jumps. Exactly one minute after Bob jumps, Charlie grabs another clock and jumps. They follow the same trajectory into the black hole, so as they each reach a given position, Charlie’s clock will show a time exactly one minute later than Bob’s clock showed at that position. Bob calculates that he will cross the event horizon exactly at 12:00 when his clock chimes midnight. Charlie’s clock will show 12:01 when he crosses the event horizon himself.

As Bob and Charlie fall, they see the apparent event horizon receding in front of them. Bob turns round and sees both Charlie behind him and Alice in the far distance. He feels a sudden pang of regret. But just then his clock chimes midnight. He has just crossed inside the event horizon and can no longer send a message to Alice. Fortunately, he can still see Charlie who is falling in one minute behind him. So he shouts to Charlie: “Tell Alice I love her”. Bob has calculated that it will take less than one minute for his message to reach Charlie, so Charlie should still be outside the event horizon and able to relay the message to Alice.

Unfortunately, Bob’s clock is running quite a bit slower than Charlie’s because Bob is at a deeper gravitational potential. According to Bob’s clock, Charlie received the message just before 12:01. But according to Charlie’s clock it is already just after 12:01. It's too late, Charlie has already crossed the event horizon. Alice will never know.