Linearly ordered rings which are not $o$-epimorphic images of ordered free rings

A proof is given that not every linearly ordered associative (associative-commutative) ring is the $o$-image of a free associative (associative-commutative) ring for some ordering of the latter. There are also nilpotent linearly ordered rings which are not $o$-epimorphic images of free associative or free associative-commutative $n$-nilpotent rings for $n\ge4$, no matter what ordering is used for the latter.