I've been trying to figure out how to properly put my confusion into a coherent thought and MrDeathTrout has brought up two great points that help me do that. It's not my intention to attack his ideas, but they do give me a great stepping point to illustrate where my confusion comes from at least.
So first off, he brings up the consensus from his previous question and that makes total sense. 'After failing' would allow you to complete the current action and then Sloth's trigger would take effect. Cool, seems to be perfectly reasonable.
Now, if we look at the example with Taelor's Crushing Blow. Here 'After Succeeding' could only mean directly after succeeding on the duel, before going to do the damage flip or anything else. And I agree with this, definitely RAI. So if we extrapolate this to go with the idea that 'After succeeding' means directly after the duel, we come to my first point of confusion: 'After Succeeding' and 'After Failing' seem to be the same tense, so one could assume that they take place at the same time. So if we take this as a basis for our logic, then Sloth's trigger would stop the action right after failing the duel...so no damage flip, nothing. This contradicts with the first point, that the consensus was that the action is resolved and then Sloth's trigger kicks in.
Okay, so directly applying the same logic as we do with Taelor's trigger may not be the correct route. Maybe we need to look at these triggers on a case by case basis, 'what makes sense to happen' RAI. So, with Crushing Blow, by this logic, we look at it and say 'Oh yea, ignoring Armor and Hard to Kill needs to be applied before damage flips, otherwise it won't do anything.' This is pretty easy to reason out. So, only looking at Sloth the argument could be made that 'hey, this attack finishes resolving, then the activation ends' just like the consensus from your previous question. And that seems totally logical also, the action fails, so after failing would take place after the failed action resolves.
Now you have two points of view, the activation is ended right after failing the duel, with the other being that the activation is ended after resolving the action. Both make sense and could probably be influenced by which side of the combat you are on
To me this is what the issue with Sloth's trigger is: that we don't have either a hard timing ruling which to apply RAW nor do we have a message from any designer to tell use what the intention of Sloth's trigger is in order to apply RAI. We would really only need one of these resolutions to be able to consistently apply Sloth's trigger.