So we are struggling since a week, 6 external people tried to make it work (server guys, wordpress experts,...). The problem doesn't seem to come from Wordpress or .htaccess file but before, however we know that if you trying to access any other page it will work. Also if a cookie is saved, the infinite loop is gone.
The .htaccess file just in case is:
<IfModule mod_rewrite.c>
Options +FollowSymlinks
RewriteEngine On
RewriteBase /
RewriteRule ^index\.php$ - [L]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule . /index.php [L]
</IfModule>
## EXPIRES CACHING ##
<IfModule mod_expires.c>
ExpiresActive on
ExpiresDefault "access plus 1 month"
ExpiresByType image/x-icon "access plus 1 year"
ExpiresByType image/gif "access plus 1 month"
ExpiresByType image/png "access plus 1 month"
ExpiresByType image/jpg "access plus 1 month"
ExpiresByType image/jpeg "access plus 1 month"
ExpiresByType video/mp4 "access plus 1 month"
ExpiresByType text/x-component "access plus 1 month"
ExpiresByType font/truetype "access plus 1 month"
ExpiresByType font/opentype "access plus 1 month"
ExpiresByType application/x-font-woff "access plus 1 month"
ExpiresByType application/vnd.ms-fontobject "access plus 1 month"
ExpiresByType text/css "access plus 1 year"
ExpiresByType application/javascript "access plus 1 year"
ExpiresByType text/javascript "access plus 1 year"
ExpiresByType text/js "access plus 1 year"
ExpiresByType application/x-javascript "access plus 1 year"
</IfModule>
## EXPIRES CACHING ##
# BEGIN GZIP COMPRESSION
<IfModule mod_gzip.c>
mod_gzip_on Yes
mod_gzip_dechunk Yes
mod_gzip_item_include file \.(html?|txt|css|js|php|pl)$
mod_gzip_item_include handler ^cgi-script$
mod_gzip_item_include mime ^text/.*
mod_gzip_item_include mime ^application/x-javascript.*
mod_gzip_item_exclude mime ^image/.*
mod_gzip_item_exclude rspheader ^Content-Encoding:.*gzip.*
</IfModule>
# END GZIP COMPRESSION
# BEGIN DEFLATE COMPRESSION
<IfModule mod_filter.c>
AddOutputFilterByType DEFLATE "application/atom+xml" \
"application/javascript" \
"application/json" \
"application/ld+json" \
"application/manifest+json" \
"application/rdf+xml" \
"application/rss+xml" \
"application/schema+json" \
"application/vnd.geo+json" \
"application/vnd.ms-fontobject" \
"application/x-font-ttf" \
"application/x-javascript" \
"application/x-web-app-manifest+json" \
"application/xhtml+xml" \
"application/xml" \
"font/eot" \
"font/opentype" \
"image/bmp" \
"image/svg+xml" \
"image/vnd.microsoft.icon" \
"image/x-icon" \
"text/cache-manifest" \
"text/css" \
"text/html" \
"text/javascript" \
"text/plain" \
"text/vcard" \
"text/vnd.rim.location.xloc" \
"text/vtt" \
"text/x-component" \
"text/x-cross-domain-policy" \
"text/xml"
</IfModule>
<IfModule mod_headers.c>
Header always set Strict-Transport-Security 'max-age=31536000' env=HTTPS
</IfModule>
# END rlrssslReallySimpleSSL
Hats off to you if you know where the issue comes from...