Conservative extensions of models with weakly o-minimal theories

Let $M\prec N$. It is said that a pair of models $(M,N)$ is conservative pair and $N$ is conservative extension of $M$ if for any finite tuple of elements $\overline{\alpha}$ from $N$, $\mathrm{tp}(\overline{\alpha}|M)$ is definable. We say that elementary extension $N$ of $M$ is $D$-good if any definable $q\in S(M\cup\overline{\alpha})$ ($\overline{\alpha}\in N\setminus M$) is realized in $N$ and $N$ is $CD$-good if any non-isolated one-type $q\in S_1(M\cup\overline{\alpha})$ ($\overline{\alpha}\in N\setminus M$), which is determined (approximated) by definable $\phi$-type, is realized in $N$.
We prove that any model $M$ of any weakly o-minimal theory except one, theory of discrete order with ends, has conservative extension. The central point in our paper is the criterion of the existence of the $CD$-$\omega$-saturated conservative extension of an arbitrary model of weakly o-minimal theory (Theorem 2). As corollary of this proof it follows the existence of $CD$-$\omega$-saturated conservative extension for any model of any weakly o-minimal theory except one and the results on omitting of natural family of definable one-types and all non-definable types (Corollary 5). The existence of conservative and $CD$-$\omega$-saturated conservative extensions for o-minimal theories have been proved accordingly in D. Marker, “Omitting types in o-minimal theories”, The Journal of Symbolic Logic, Vol. 51(1986), P. 63–74., Y. Baisalov, B. Poizat, “Paires de structures o-minimales”, The Journal of Symbolic Logic, Vol. 63(1998), P. 570–578.