  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. 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 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,
these define a thin set which coincides with the conjectural exceptional set.
