12.51 (1992)

Does the free group $F_\eta$ ($1 < \eta < \infty$) have a finite subset $S$ for which there is a unique total order of $F_\eta$ making all elements of $S$ positive? Equivalently, does the free representable $l$-group of rank $\eta$ have a basic element? (The analogs for right orders of $F_\eta$ and free $l$-groups have negative answers.)