Fixed-point conjecture for L-space knots

Let $K$ be an L-space knot, and let $\mu$ and $\lambda$ represent a meridian and a longitude in the knot group. For a homomorphism $\rho$ from $\pi_1(S^3\setminus K)$ to $\operatorname{Homeo}_+(\mathbf{R})$, say that it has no global fixed point when the action has no point fixed by the whole image. Fixed-point conjecture for L-space knots. There exists an element $k$ in the knot group such that, for every homomorphism $\rho$ from $\pi_1(S^3\setminus K)$ to $\operatorname{Homeo}_+(\mathbf{R})$ without global fixed points, if $\rho(\mu)$ and $\rho(\lambda)$ have the same set of fixed points, then $\rho(k)$ has no fixed points. The statement is proposed as a possible key step toward proving that fundamental groups of L-spaces obtained by Dehn surgery are not left orderable; the source gives no resolution status.