Google does not support WebSocket connections in Googlebot. This was first known, I think, back in May 2019, when Martin Splitt and Zoe Clifford presented at Google I/O. Zoe Clifford said there "Google does not allow WebSocket connections in Googlebot for architectural reasons."
Martin Splitt was recently asked if this is still true and Martin replied "As far as I know, not available as of today, no."
WebSocket is a computer communications protocol, providing full-duplex communication channels over a single TCP connection. The WebSocket protocol was standardized by the IETF as RFC 6455 in 2011. The current specification is known as the HTML Living Standard. It is maintained by the Web Hypertext Application Technology Working Group (WHATWG), a consortium of the major browser vendors (Apple, Google, Mozilla, and Microsoft).
This is the code sample Martin and Zoe shared in the Google I/O talk:
Here are those new tweets:
As far as I know, not available as of today, no.
— Martin Splitt (@g33konaut) April 10, 2022
Forum discussion at Twitter.