EPiServer, EPiServer.SchedulerSvc version 5.2.375.236

In CMS 5 the Scheduler run jobs by calling the site and then using the API/objectmodel.
I have a job scheduled every night at 3.00. If I enable "Anonymous access" in the IIS it works OK. But if I disable "Anonymous acccess" in the IIS it does not run before someone browses the site, normally sometime between 8.00 and 9.00.

The Scheduler log sais (Anonymous access disabled):
[2010-06-01 19:34:53UTC] #INF# [_LM_W3SVC_207710262_root] Site down for 10+ minutes, open http://mysiteurl/
[2010-06-01 19:35:53UTC] #INF# [_LM_W3SVC_207710262_root] Failed calling site (Attempting to deserialize an empty stream.)
[2010-06-01 19:36:53UTC] #INF# [_LM_W3SVC_207710262_root] Failed calling site (Attempting to deserialize an empty stream.)
etc............................................

The IIS log sais (Anonymous access disabled):
401 2

By using Reflector I have tracked the problem to the "EPiServer.SchedulerService.JumpStartWEB.StartSite(string sUrl)" method. This method do:
WebRequest state = WebRequest.Create(sUrl);
state.BeginGetResponse(new AsyncCallback(JumpStartWEB.dontCare), state);

No credentials are set.

I have a little test application where I execute:
WebRequest state = WebRequest.Create("http://mysiteurl");
WebResponse response = state.GetResponse();

This throws an exception:
System.Net.WebException : "The remote server returned an error: (401) Unauthorized."

If I enable anonymous access my code works ok.

So my conclusion is that when the EPiServer.SchedulerSvc.exe tries to wake the site it can't because it is unauthorized.

My question: Can the EPiServer Scheduler handle EPiServer sites where "Anonymous access" is disabled in the IIS? If not, how should I work around it?

/Hans