No. That is not what this says. For entry to the USVI, either by air or water, proof of vaccination uploaded to the USVI portal is all that is required. No tests of vaccinated.
However, BVI still requires a negative test for admission. Hopefully that will change to be consistent with the USVI. Most likely dependent on hospitalization usage in the BVI.
Entry to USVI from BVI by air will still be subject to CDC requirements for international air arrivals. USVI is the first port of air entry. See page 9 item C.
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