It looks like their API only supports ETH and the docs say that it's in alpha, so we may see it supporting EOS soon. You can ask in their Telegram to see if the API is ahead of their docs.
With that said, they have a separate repo for their eos contract. From the readme:
BancorNetwork is the entry point for any token to any token
conversion. BancorConverter is responsible for converting between a
specific token and its own reserves.
In order to execute a conversion, the caller needs to transfer tokens
to the BancorNetwork contract with specific conversion instructions in
the transfer memo.
Here's the description for the eos converter contract and the link to it:
The Bancor converter allows conversions between a smart token and tokens
that are defined as its reserves and between the different reserves directly.
Reserve balance can be virtual, meaning that the calculations are based on
the virtual balance instead of relying on the actual reserve balance.
This is a security mechanism that prevents the need to keep a very large
(and valuable) balance in a single contract.