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The Hunt for Observable Signatures of Terrestrial Systems survey searches for dust near the habitable zones (HZs) around nearby, bright main-sequence stars. We use nulling interferometry in the N band to suppress the bright stellar light and to probe for low levels of HZ dust around the 30 stars observed so far. Our overall detection rate is 18%, including four new detections, among which are the first three around Sun-like stars and the first two around stars without any previously known circumstellar dust. The inferred occurrence rates are comparable for early-type and Sun-like stars, but decrease from 60(sup +16)(sub -21)% for stars with previously detected cold dust to 8(sup +10)(sub -3)% for stars without such excess, confirming earlier results at higher sensitivity. For completed observations on individual stars, our sensitivity is five to ten times better than previous results. Assuming a lognormal excess luminosity function, we put upper limits on the median HZ dust level of 13 zodis (95% confidence) for a sample of stars without cold dust and of 26 zodis when focusing on Sun-like stars without cold dust. However, our data suggest that a more complex luminosity function may be more appropriate. For stars without detectable Large Binocular Telescope Interferometer (LBTI) excess, our upper limits are almost reduced by a factor of two, demonstrating the strength of LBTI target vetting for future exo-Earth imaging missions. Our statistics are limited so far, and extending the survey is critical to informing the design of future exo-Earth imaging surveys.