Let $S^3-K$ be the complement of the figure eight knot complement. Thurston, in his Lecture Notes, constructed a hyperbolic structure, which comes from a discrete, faithful representation $\pi_1(S^3-K)\to SL(2,{\mathcal O}_3)\subset SL(2,C)$. This representation is not conjugate into $SU(2)$, because traces are not real. But, of course, one can get nontrivial $SU(2)$-representations factoring over the abelianization $Z$ of $\pi_1(S^3-K)$.

Thurston goes on to prove that almost all Dehn surgeries at the figure eight knot complement admit a hyperbolic structure, hence a faithful, discrete representation $\pi_1(S^3-K)\to PSL(2,C)$, which by Culler lifts to $SL(2,C)$ and again is not conjugate into $SU(2)$.

On the other hand, Kronheimer and Mrowka show that for Dehn surgery coefficients $\vert\frac{p}{q}\vert\le 2$, the Dehn surgered manifold admits a noncyclic representation to $SU(2)$. This is, in a sense, complementary to Thurston‘s result, who proved his result for sufficiently large Dehn surgery coefficients.

Question: is it known whether Dehn surgeries at the figure eight knot complement, with sufficiently large Dehn surgery coefficients, admit nontrivial (not necessarily faithful) representations to $SU(2)$?