Calcium concentration is by far the largest factor. If an oil is low enough in calcium (say, <1000-1200 ppm), it will generally perform well on an LSPI test, regardless of what's in the rest of the formulation.
Oils with moderately high calcium (say, around 1200-1500 ppm) can still sometimes perform well enough in LSPI tests if they contain high concentrations of moly and/or ZDDP, but it takes a lot of these additives to make up for a modest increase in calcium. Within the 600-800 ppm phosphorus range required for API SP oils, there won't be much difference in LSPI reduction from the ZDDP.
Oils with very high calcium (say, >2000 ppm) will not pass an LSPI test regardless of how much moly or ZDDP is added to the oil, at reasonable concentrations.
HPL's passenger car oils tend to be in the 900-1200 ppm Ca range as far as I've seen, so I'd expect them to perform well, even if they aren't actually subject to the API or Dexos tests.