Generally yes, but if you use e.g. the parallel consumer, you can potentially keep processing in that partition to avoid head-of-line blocking. There are some downsides to having a very old unprocessed record since it won't advance the consumer group's offset past that record, and it instead keeps track of the individual offsets it has completed beyond it, so you don't want to be in that state indefinitely, but you hope your DLQ eventually succeeds.

But if your DLQ is overloaded, you probably want to slow down or stop since sending a large fraction of your traffic to DLQ is counter productive. E.g. if you are sending 100% of messages to DLQ due to a bug, you should stop processing, fix the bug, and then resume from your normal queue.