I have a WordPress site running on OpenLiteSpeed, and accessing XMLRPC with cURL like:

curl --data @hello.txt https://gateblogs.com/xmlrpc.php

with the correct info in hello.txt, returns:

<?xml version="1.0" encoding="UTF-8"?>
<methodResponse>
  <params>
    <param>
      <value>
      <string>Hello!</string>
      </value>
    </param>
  </params>
</methodResponse>

but, Jetpack Debugger gives me a Communication error. I don't get this because XMLRPC obviously works, and Jetpack worked with Apache, so I know none of my plugins are blocking it.

I also used:

curl -A 'Jetpack by WordPress.com' -is -H 'Content-Type: text/xml' --data '<?xml version="1.0"?><methodCall><methodName>demo.sayHello</methodName><params></params></methodCall>' 'https://gateblogs.com/xmlrpc.php' && echo

Apache(what works) returned:
0; path=COOKIEPATH
Set-Cookie: QnKVvgyjBb=e4x5FOl1S%40; expires=Wed, 28-Mar-2018 15:06:28 GMT; Max-Age=3600; path=COOKIEPATH
X-LiteSpeed-Cache-Control: no-cache
Vary: Accept-Encoding
Expect-CT: max-age=604800, report-uri="https://report-uri.cloudflare.com/cdn-cgi/beacon/expect-ct"
Server: cloudflare
CF-RAY: 402aa831dc789248-EWR

<?xml version="1.0" encoding="UTF-8"?>
<methodResponse>
<params>
<param>
<value>
<string>Hello!</string>
</value>
</param>
</params>
</methodResponse>

Litespeed returned:
Set-Cookie: c_uINfQa=FEsjKuzn; expires=Wed, 28-Mar-2018 15:07:35 GMT; Max-Age=3600; path=COOKIEPATH
Set-Cookie: QnKVvgyjBb=e4x5FOl1S%40; expires=Wed, 28-Mar-2018 15:07:35 GMT; Max-Age=3600; path=COOKIEPATH
X-Litespeed-Cache-Control: no-cache
Expect-CT: max-age=604800, report-uri="https://report-uri.cloudflare.com/cdn-cgi/beacon/expect-ct"
Server: cloudflare
CF-RAY: 402aa9efd9a791a0-EWR

<?xml version="1.0" encoding="UTF-8"?>
<methodResponse>
<params>
<param>
<value>
<string>Hello!</string>
</value>
</param>
</params>
</methodResponse>

The only difference I see is the second Set-Cookie for Litespeed