Metric-measure and Heisenberg-group extension of the Neumann problem result

Let

$$denote the paper's Neumann problem with discontinuous nonlinearity, posed in the Euclidean domain setting. The proposed extensions replace that setting by either a metric measure space or a Heisenberg group. **Metric-measure and Heisenberg-group conjecture.** Problem$$

should yield the same result when considered over a metric measure space or a Heisenberg group.

This conjecture asks whether the corresponding conclusion for the Euclidean problem persists in these two non-Euclidean frameworks. The supplied text gives no resolution or precise structural assumptions on the metric measure space or Heisenberg group.