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Yes, the LSP supports unannounced/private channels. If you set announce_channel: false in the channel request, the LSP will open an unannounced channel with you.
{
    "node_connection_info": "string",
    "lsp_balance_sats": 50000000,
    "client_balance_sats": 0,
    "confirms_within_blocks": 6,
    "channel_expiry_blocks": 12288,
>>  "announce_channel": false
}