The geometric exceptional set in Manin's conjecture for Batyrev and Tschinkel's example

Batyrev and Tschinkel's example is a Fermat cubic surface bundle $X$ which is a Fano $5$-fold. It is the first example for which Manin's conjecture can never hold for a proper closed exceptional set. Recently, Lehmann, Sengupta, and Tanimoto proposed a conjectural geometric description of the exceptional set in Manin's conjecture and showed that it is always contained in a thin set. Over a field of characteristic $0$, we explicitly construct finitely many thin maps such that any thin map $f:Y\rightarrow X$ with equal or larger $a$- and $b$-values in lexicographical order factors rationally through one of them. In particular, this defines a thin set which coincides with Lehmann-Sengupta-Tanimoto's conjectural exceptional set.