I haven't been able to reproduce the issue you mentioned about favicon.ico. For example, i just visited a site and checked request headers sent, & everything seems functioning OK:
Host: static.bfmtv.com
User-Agent: Mozilla/5.0 (Windows NT 6.1; Win64; x64; rv:64.0) Gecko/20100101 Firefox/64.0
Accept: text/html,application/xhtml+xml,application/xml;q=0.9,*/*;q=0.8
Accept-Language: fr,fr-FR;q=0.8,en-US;q=0.5,en;q=0.3
Accept-Encoding: gzip, deflate, br
DNT: 1
Connection: keep-alive
Cookie: TCPID=119131139489067326342; xtvrn=$548539$; xtan=-; xtant=1; etuix=9cc_5.jv5oCi69wkDT9jtGVIJNVLip69oIcOR5ViDZO_4giNojVnug--; ownpage_fp2=7ebfab6931223c16
Pragma: no-cache
Cache-Control: no-cache
TE: Trailers