Conjecture that the quantum measure determines the harmonic extension off the set

Let $X$ be a local set coupled with a Gaussian free field $h$, including a random set independent of $h$, and let $cmu_{X,h}$ be the associated quantum measure. The harmonic extension of $h$ off $X$ is the natural interpretation of the field restricted to $X$.

Quantum-measure determination conjecture. In the setting described above, the measure $cmu_{X,h}$ almost surely determines the harmonic extension of $h$ off $X$.

This is a precise formulation of the question of how much information the quantum measure carries about the field. The supplied text gives no resolution status beyond presenting it as a conjecture.