The critical-value conjecture for the one-dimensional continuum random-cluster model

Let ${\mathbb P}_{\lambda,\delta,q}$ be the continuum random-cluster measure on ${\mathbb Z}\times{\mathbb R}$, let $\theta(\lambda,\delta,q)$ be the probability that the cluster of the origin is infinite, and define

$$\lambda_{\mathrm c}({\mathbb Z}^d,q)=\sup\{\lambda:\theta(\lambda,1,q)=0\}.$$

The one-dimensional continuum random-cluster critical-value conjecture. For $q\geq1$,

$$\lambda_{\mathrm c}({\mathbb Z},q)=q.$$

This conjecture is motivated by the self-duality of the one-dimensional random-cluster model; the source gives no resolution, so its status remains open.