Singularity conjecture for the first exceptional time infinite cluster

Let $\mathsf{FETIC}$ be the law of the configuration at the first exceptional time, and let $\mathsf{IIC}$ denote the incipient infinite cluster measure. Singularity conjecture. The measures $\mathsf{FETIC}$ and $\mathsf{IIC}$ are singular with respect to each other. This would strengthen the known fact that these two measures are not equal, and would clarify how the first exceptional time differs from a typical exceptional time. The source provides no resolution of this conjecture.