Can you capture the last piece in chess?

RIGHT and WRONG. Resigning after DP is probably not normal as the game has already finished even when the players and the arbiter don't know about it. I saw this clarification coming before the turn of the century and was pleasantly surprised to hear from @jetoba last year that it did indeed crystallize in almost the exact way I had predicted.

If a player resigns in a dead position, without either realizing it's a dead position, the opponent accepts it as a loss, and they report it to the arbiter without the arbiter seeing the endgame position, then that's where I don't think the results would be changed. Maybe FIDE does allow results to be changed later, after being reported and entered.

Though, I wonder how often it happens in practice. My guess is that it's not often and mostly at lower rating bands

Apparently it happens often enough for the referee support team to provide the clarification. We didn't have to start a new query for it, it was already in the database. You'll be surprised to read how far they would go to correct the wrong end result. Almost anything goes (including undoing a signed scoresheet if I recall well) as long as the event organization is not unduly disrupted. I'll send you the link if I find it. The information is public.

Btw, the composers now have a computer program that can test for dead positions. With an implementation for the game environment - and obviously consensus on the definition of dead positions - it could report any missed dead positions during the game and rectify the end result in seconds or minutes after the game is over!

Here is the FIDE-link about 75M, 5rep and Resignations in draw positions. the answers to the first 2 questions apply, the 3rd one is on a different subject.

https://rcc.fide.com/2021/11/08/questions-answers-november-2021/

The text is not as clear as we would like. There are mysterious phrases like "The game is over when a mate, stalemate, 5-fold repetition, or 75 moves happen and the further moves are not counted. Which would imply the moves until the 75M line were counted - then the game was continued though it was known that the 75M line was reached - then the counting was discontinued instead of drawing the game - to make sure the DP rule would never kick in? Penalized for counting? Total madness!

OK, results can be changed later if warranted. So that answers that part of the discussion but not the main one from the topic.

Correct. The main one is the question whether or not we still have a functioning DP rule in chess today. Based on the reply jetoba received on his query - not the one above - we would have to believe that no chess position is dead including the one in the OP (with or without knight). To keep DP you'd need to accept the analysis of my posted diagram - and many others like MARattigan's near-75M construction used for jetoba's query.

So, nothing is resolved yet - but I would challenge any arbiter in court declaring any game position dead to enforce a substantiated ruling

It's still clear to me that it's pointless to play out positions that are covered by 5.2.2, that would result in a draw on timeout or otherwise would end in one of the other draw conditions (no other options)

I'll probably never be a FIDE arbiter (FIDE rated events is my area wouldn't be very workable for the most part anyway) but unless there is a new rule published, I would certainly rule such a position a draw and would expect the vast majority of players would agree.

To be precise, the Dead Position rule always was only there to convert that uneasy feeling of "continue playing this position makes no sense whatsoever" into a law allowing the arbiter to intervene under defined conditions. You might say that effort failed considering the presented cases. So it stands to reason to withdraw the DP rule as it stands. That is the preference of @jetoba expressed in the discussion. Which permits the players to troll around unchecked. Btw, that even happens now with 5R and especially 75M as is already demonstrated in many pro-games. Which is natural since ignoring these marks still leaves the checkmate opportunities intact even when an actual checkmate should be overridden by the arbiter just as resignation and the players could claim at any time before the checkmate!

There is another factor of interest though you probably don't care a lot about it. The composition world uses the DP definition "by the letter" - simply because it has no DP definition of its own. Most rules in orthodox composition chess are inherited from FIDE by default. Whatever FIDE changes now will likely create havoc in the validation of its orthodox retro-problems. Fortunately DP has been disabled for the most orthodox problem types but the parting of the American and European continents 100.000.000 years ago also started with a 1 inch gap. Since many of the composed puzzles on chess.com come from the composition domain they may increasingly start to operate under different rules. Certainly some of my own since I used DP for ages!

it should end at that exact position