Eventual equality of the Thurston and higher-order Alexander norms

Let $M$ be a $3$-manifold, let $\pi_1(M)_{r}^{(n)}$ denote the $n$th term of its rational derived series, and define

$$\pi_1(M)_{r}^{(\omega)}=\bigcap_{n\in\mathbb{N}}\pi_1(M)_{r}^{(n)}.$$

For each $n\in\mathbb{N}$, let $||\\,\cdot\\,||_n$ be the corresponding higher-order Alexander seminorm, and let $||\\,\cdot\\,||_T$ be the Thurston seminorm. Eventual equality conjecture. If

$$\pi_1(M)_{r}^{(\omega)}=\\{1\\},$$

then there exists $n\in\mathbb{N}$ such that

$$||\\,\cdot\\,||_T=||\\,\cdot\\,||_n.$$

The conjecture asks when the sequence of higher-order Alexander seminorms eventually agrees with the Thurston seminorm. The preceding discussion gives examples where eventual agreement is known, while the general question remains open.