Anyone compared HL's native TWAP with sending the slices yourself?

Found a June paper on HL execution with a result I wasn't expecting: native TWAPs had lower price impact than comparable trades where the full order wasn't announced upfront. The book also showed more liquidity on the side absorbing the TWAP. Paper That complicates the intuition that making your schedule public just gives other traders something to trade ahead of. But the authors flag a pretty big limitation: these are mostly different traders using different methods. It isn't the same person randomly switching between public and private execution. Has anyone here compared native TWAP with their own slices on the same pairs, at roughly similar size and urgency? Did hiding the schedule make enough difference to justify running your own execution? An example where you switched back would be useful too. The average result could easily miss the thin market where being predictable really hurts