The representation-variety conjecture detecting the trefoil

Given a knot $K$ in the 3-sphere, let

$$\mathscr{R}(K,\mathbf{i})=\{\rho:\pi_1(S^3\smallsetminus K)\to SU(2)\mid \rho(\mu)=\mathbf{i}\},$$

where $\mu$ is a chosen meridian and

$$\mathbf{i}=\left[\begin{array}{cc} i&0\\ 0&-i\end{array}\right].$$

For a trefoil $T$, its representation variety is $\mathscr{R}(T,\mathbf{i})\cong\{*\}\sqcup S^1$, with $*$ the reducible homomorphism and $S^1$ the unique conjugacy class of irreducibles. The representation-variety conjecture.

$$\mathscr{R}(K,\mathbf{i})\cong\{*\}\sqcup S^1$$

if and only if $K$ is a trefoil. This proposes that the specified $SU(2)$ representation variety characterizes the trefoil among knots.