No you just on-chain transfer into cold storage. LN is what you use on the way out of cold storage, but not through swap but you simply fund a channel. Boltz does mixing checks so you only want to use them to offramp from LN. For onramp, just find a cheap counterparty on Amboss.

Edit: My last round of mixing (at 2-5 sat/vb) cost me ~0.3% in fees in total. So 0.5% for boltz + 0.3% for mixing = 0.8%... LN fees plus - unless you're unlucky - tx fees out, won't add .2% there if you're using Msat denominations... so under 1%.