No. There is no official conversion table, and the vocabulary loads do not line up: old HSK 3 expected 600 words while new Band 3 expects far more. As a rough map, old HSK 4 sits around new Band 3 to 4 and old HSK 5 around Band 4 to 5.

Around Band 5 to 6 in raw vocabulary, not Band 9. Old HSK 6 was the ceiling at 5,000 words; the new Advanced stage (Bands 7–9) describes competence the old test never measured.

Old HSK certificates remain valid for their stated validity window — a transcript saying HSK 5 still means what it meant. If a program lists a requirement in the old format, your old certificate likely satisfies it.

Benchmark rather than convert. Take a placement check against the old HSK 1–6 scale, then pull up a word list one band above where you think you are and see how much is genuinely new — and test all four skills, since the new standard weights speaking and writing far more heavily.