Dion–Ray conjecture on vanishing Iwasawa invariants for 2-bridge links

Let $p$ be a fixed odd prime. Let $\mathcal{N}$ be the set of fractions indexing $2$-bridge links, and for $a/b\in\mathcal{N}$ let $\mu_{p,a/b,\mathbf{1}}$ denote the associated topological Iwasawa $\mu$-invariant. Define

$$\mathcal{U}^p=\left\{a/b\in\mathcal{N}:\mu_{p,a/b,\mathbf{1}}=0\right\}$$

and, for $\mathcal{U}^p_{<x}$ and $\mathcal{N}_{<x}$ consisting of fractions of height less than $x$, define

$$\mathfrak{d}(\mathcal{U}^p):=\lim_{x\rightarrow\infty}\frac{\#\mathcal{U}^p_{<x}}{\#\mathcal{N}_{<x}}.$$

Dion–Ray conjecture. The set $\mathcal{U}^p$ has density

$$\mathfrak{d}(\mathcal{U}^p)=1.$$

The conjecture asserts that, for every fixed odd prime, the topological $\mu$-invariant vanishes for almost all $2$-bridge links. This refines the paper's unconditional density result for the simultaneous conditions $\mu=0$ and $\lambda=1$, and is motivated by computational evidence; individual counterexamples show that vanishing need not hold for every link.