What kernel to use for Gaussian Process Regression on data with large flat region?

Question

The problem I'm working on has a region that's largely accurately modeled by a GP model using a squared exponential kernel. However, there is a large region in the "truth" model that is uniformly zero, with all samples taken from this region being zero as well. However, the GP model in this region oscillates from positive to negative when it's necessary for the problem I'm working on that this region is accurately modeled as uniformly zero, or at least negative when the remainder of the data is positive. Adding a dot product kernel to the squared exponential kernel to create a custom kernel helps somewhat but can't eliminate the spurious oscillations.