Instanton foliation-depth conjecture for knots

Let $K\subset S^{3}$ be a knot. Consider the irreducible homomorphisms $c1:c0_{1}(S^{3}\setminus K)\to\operatorname{SU}(2)$ that map a chosen meridian $m$ to the element $bfi\in\operatorname{SU}(2)$. Suppose these homomorphisms are non-degenerate and that the number of their conjugacy classes is less than $2^{k+1}$. Instanton foliation-depth conjecture. The knot complement $S^{3}\setminus N^{\circ}(K)$ admits a foliation of depth at most $2k$, transverse to the torus boundary. Motivated by Juhász's extension of the fibering theorem, this conjecture predicts a bound on foliation depth from the number of non-degenerate instanton representations; the source gives no resolution.