Connection to API drops during SSL handshake

I'm supporting a customer who is beginning to try our product's integration with the ecobee API. (I believe they're currently testing against beta.) They are seeing intermittent communication errors. The connection to ecobee seems to be failing before the request is sent, during the SSL handshake.

SSLHandshakeException: Remote host closed connection during handshake
EOFException: SSL peer shut down incorrectly

This happens on multiple types of requests, including POST to /hierarchy/set and POST to /hierarchy/thermostat (register and move). But it does not happen consistently.

I'm wondering if ecobee has seen connections fail like this before and might have an idea how best to troubleshoot.
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