Kronheimer–Mrowka's depth bound conjecture for knot complements

Let $K\subset S^3$ be a knot, and let $S^3(K)$ denote its knot complement. Consider the irreducible homomorphisms

$$\rho:\pi_1(S^3(K))\longrightarrow SU(2)$$

that map a chosen meridian $m$ to $\mathbf{i}\in SU(2)$. Assume these homomorphisms are non-degenerate, and that the number of their conjugacy classes is less than $2^{k+1}$. Kronheimer and Mrowka's conjecture. The knot complement $S^3(K)$ admits a taut foliation of depth at most $2k$, transverse to its boundary $\partial S^3(K)$.

This conjecture is an instanton-theoretic analogue of Juhász's proposed depth bound for taut foliations. The supplied source gives no resolution or proof status beyond stating the conjecture.