We where for a long time interested in connecting OpenBCI to Home Assistant, for joining sensor data with sensor data from other devices in InfluxDB.
MQTT over WIFI shield: we know of someone who tried it, but did not confirm that it works
OpenBCI MQTT implemented in a paper from authors from Equador: Connect to the authors, if you are interested in coders which can do this.
Python LSL requires an x86 processor
we do not know of a straight-forward pathway from OpenBCI to Home Assistant
For us personally, it seems easier to conned single EEG modules to an 16 bit or higher A/D converter to ESP32 modules which can be connected easily to Home Assistant
OpenBCI offers multi-sensor support up to 16 channels
Home Assistant with InfluxDB also offers multi sensor support, therefore it does not require devices with multi-sensor support
Physiological sensors are poorly implemented in Home Assistant to this date.