Khoussainov's conjectures on quotient presentations of nonstandard arithmetic models

A computable quotient presentation of a mathematical structure $\mathcal A$ consists of a computable structure on the natural numbers $\mathbb N$, together with an equivalence relation $E$ on $\mathbb N$ that is a congruence, such that the quotient is isomorphic to $\mathcal A$. An equivalence relation is computably enumerable if its pairs can be enumerated by a computable procedure, and co-c.e. if its complement is computably enumerable.

Khoussainov's conjectures. (1) No nonstandard model of arithmetic admits a computable quotient presentation by a computably enumerable equivalence relation on the natural numbers. (2) Some nonstandard model of arithmetic admits a computable quotient presentation by a co-c.e. equivalence relation.

The conjectures concern the possible quotient presentations of nonstandard models of arithmetic and form part of a program using computable quotient presentations in computable model theory. The source states that the first conjecture is proved, several natural variations of the second are refuted, and a further natural variation remains open; it does not establish either listed conjecture as resolved in the supplied passage.