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Re:I exploited TLS-SNI-01 issuing Let's Encrypt SSL-certs fo... by schoen on Tuesday January 29, 2019 @10:11PM (#19032303)
This thread seems to agree with you:

https://stackoverflow.com/questions/1612177/are-http-cookies...

I just tested this and I was able to confirm your suggestion with Firefox, watching with a packet sniffer while connecting to my own site with http://example.com:443/ after previously setting a cookie over https://example.com/. The cookie was clearly visible in the packet log.

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