Is the $S$matrix conjecture still open? I mean the one listed as Problem 7 in this survey.
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To my knowledge, this problem is still open. There has been partial progress on it in the past few years, but it still seems quite far from resolved. The latest paper that I am aware of is: here, though unfortunately behind a pay wall. 


Here is a more recent paper: Roman Drnovšek, On the Smatrix conjecture. It (re) proves for the case of $n$ odd the conjecture restated here for convenience: 

