OK. Events by Timeline:
Joseph claims Neo Cop, claiming to target Joth and getting a not VT result.
Joth claims JK.
Joth then says they targeted ADK night 1 and Swan night 2.
We then had no Night Kill happen.
So what are the possible things that could be happening:
1) They could both be telling the truth - Joseph got the Neo shot, used it on Joth. Joth JKd Swan. Skum tried to kill Swan.
2) Joseph is skum, Joth is JK - Skum was informed that the Neo shot (or anything really) was to be assigned, but they got the info instead. Joseph claims on Joth, and just happens to accidentally pick the player that is actually the JK.
3) Joseph is town, Joth is skum - Joseph got the neo shot ability, used it on Joth. Joth claims JK as skum hoping to out the real JK in the process.
4) They are both skum - Skum gets the info of the power. Joseph claims neo on Joth. They duke it out hoping to pull out the JK. The JK assumedly claims, we lynch Joth, Joseph gets town points X 1000. If JK doesn't claim, it still works. No matter who is lynched, the other person looks good.
---1) I think this is very unlikely. It is mainly based on just pure chance of shit happening, but I mean that is just like the stars aligning.
---2) This could be a thing. Skum gets the info of the power, picks a player that they think is safe (Joth would be prime as they are the only player alive that would of had any reason at all to claim JK at any point), but then it backfires. It also has the added benefit for skum of potentially having a quick JK claim come out. Like Joseph says Joth is not VT, then JK comes out and says "then they are for sure skum, because I am the JK".
---3) This could also be a thing. Joseph gets the shot, uses it, hits skum... skum acts accordingly.
---4) This is like #1 in far off worldness... but should not be ignored. At this stage, especially after the kill was blocked, skum could be worried the JK block will be an issue as the field lessens. This set up would ensure one skum will get lynched, but it also gives a pretty good chance that the real JK comes out and then they have that player at night.