https://www.sciencedirect.com/science/article/pii/S0166218X19301969

http://www.math.kit.edu/iag6/~axenovich/media/path-trans.pdf

Both papers proves that the conjecture holds with certain graphs. Are there more papers like these?

3

Some works on Zamfirescu's Conjecture: In a connected graph the intersection of any three distinct longest paths is nonempty.

https://www.sciencedirect.com/science/article/pii/S0166218X19301969

http://www.math.kit.edu/iag6/~axenovich/media/path-trans.pdf

Both papers proves that the conjecture holds with certain graphs. Are there more papers like these?

The second paper mentions that this holds in the case of two longest paths. What is the intuition behind this proof?

If the two paths didn't share a vertex then you could construct a new path using the two longest paths (since the graph is connected) that is longer than the longest paths, resulting in a contradiction.