WebMux (v12 +) supports and does health checks using the Server Name Indication (SNI) TLS extension.
To ease the administration of SSL/TLS certificates, SNI is almost universally used at the present time. This extension to SSL/TLS allows specifying explicitly the domain name for which a SSL/TLS connection is requested. Thus the same server can have several separate certificates, each for a different virtual host. These certificates may have different expiration dates and be obtained from different certificate authorities. Thus if a new virtual host is added, it is not necessary to worry about any of the certificates for the existing hosts; it is only necessary to get a new certificate for the new virtual host.