Since now more and more people seem to have solved Eppstein's puzzle I am curious whether or not our approaches agree, and to what extent they do.

Therefore, hopefully without spoiling the fun too much for those who still would like to try, I will post the last chain in the bilocal graph that I had used and the penultimate one:

[STILL, I RECOMMEND TO SKIP READING FURTHER IF YOU ARE INTERESTED TO SOLVE EPPSTEIN'S PUZZLE FROM SCRATCH!]

The last chain started in box number 9 and established the number which finally "collapsed the Sudoku". It was a closed cycle with labels 521736152795 (pretty lengthy, eh?).

The penultimate chain started in box number 3 and established a key number in that box (which allowed the chain above). It led to an application of the "conflicting path rule" and the paths were 349 and 393429. The two paths overlap on one edge, but you may check easily that the method still works.

Another reason why I restrict myself to the last two chains is that, embarrassingly, I could not reconstruct how I set up the first chain (it is a bit difficult by looking at the bilocation graph, because of course it was growing during the solution, and I just have the "final" graph on a sheet of paper). I saw in a margin scribbled 776(47)8, where the notation (47) means that the edge carries the labels 4 and 7, but such a closed path does not seem to exist. I then noted that instead the closed path 776(68)8 exists, which at first glance allows to apply the closed cycle rule: after all, the only label that repeats itself is 7, while the other labels, 6, 8 and (68), are all different - but it is not true. The label (68) for all practical purposes should be considered as two edges, one with label 6 and one with label 8. But in the sequence 6(68)8 you will necessarily have a repetition, no matter which of the two ways you choose.

Thus, either there is another path at this stage which yields the crucial 7 in box 2, or my "first application of the bilocation graph technique" was actually a lucky guess.

Therefore my question to those who have solved the puzzle: which chains in the bilocation graph did you employ?