It has been proved in our earlier papers [Algebra Logika 28, No. 5, 597- 607 (1989; Zbl 0708.08005); Sib. Mat. Zh. 31, No. 3, 125-134 (1990; Zbl 0712.08008)] that 1) if \({\mathfrak M}\) is a discriminator variety that is not finitely generated, then \(\langle {\mathfrak {JM}}_{\aleph_ 0};\ll\rangle\), its countable epimorphism skeleton, contains an uncountable set of pairwise incomparable elements, and if \({\mathfrak M}\) is of finite signature or all \({\mathfrak M}\)-algebras contain one-element subalgebras, then each countable quasiorder can be isomorphically embedded in \(\langle {\mathfrak {JM}}_{\aleph_ 0};\ll\rangle\); 2) if \({\mathfrak M}\) is a discriminator variety of finite signature that is not locally finite, then each countable partially ordered set can be isomorphically embedded in \(\langle {\mathfrak {JM}}_{\aleph_ 0}; \leq\rangle\), its countable embeddability skeleton. In the same articles we have put forward the conjecture that in the case of finitely generated discriminator varieties \({\mathfrak M}\) both the countable skeletons \(\langle {\mathfrak {JM}}_{\aleph_ 0}; \ll\rangle\) and \(\langle {\mathfrak {JM}}_{\aleph_ 0}; \leq\rangle\) are totally quasiordered, i.e., each set of their incomparable elements is finite, and each strictly decreasing chain terminates. The present article is devoted to a proof of these statements.