Algebraic fibering is generic for random two-generator one-relator groups with commutator relator

Let $\mathcal{G}'(r)$ denote the random model of two-generator one-relator groups whose defining relation is a commutator, and let $p_r$ be the probability that a group $G\in\mathcal{G}'(r)$ fibers algebraically. Algebraic-fibering conjecture. One has

$$p_r\to 1\quad\text{as }r\to\infty.$$

This contrasts with tunnel number one 3-manifolds with two boundary components, whose fibering probability tends to $0$ despite having many epimorphisms to $\mathbb Z$. The source describes the assertion as very likely and gives no proof.