SqlCeReplication with client certificates

We have a lot of devices which synchronize their databases using the SQL CE Merge Replication (over SSL and with Basic Authentication configured). Now we want our clients to additionally identify themselves with a client certificate, but we're unsure, whether this configuration is supported or not.

In the web configuration wizard, you can require the clients to provide a certificate:

But there's no property on the SqlCeReplication class which allows to specify a certificate from a file or the certificate store. Therefore, the replication attempts fail with an error:

A request to send data to the computer running IIS has failed. For more information, see HRESULT. NativeError: 28037 HResult: 0x80072F0C

Currently, I'm very unsure whether client certificates are supported by the SQL CE Merge Replication or not. There are three pages on MSDN which state very cleary:

Web synchronization for merge replication supports using server certificates but not client certificates.

But why does the Web Synchronization Wizard offer that option then? I would be very pleased if someone could tell me for sure, whether client certificates are possible or not.