Access details of contact registering conversion

We're looking at using conversions to trigger integration with a CRM (not Salesforce). I can use the following code to add a custom handler after logging a conversion where I can find the site and conversion names:

public override void Init()
{
    WebAnalyticsEvents.LogConversion.After += LogConversion_After;
}

void LogConversion_After(object sender, CMSEventArgs<LogRecord> e)
{
    var siteName = e.Parameter.SiteName;
    var conversionName = e.Parameter.ObjectName;
    // TODO: Get contact details and push to CRM
}

What I'd like to know is, is it possible from here to get details of the contact who has triggered the conversion? If the contact is not anonymous I need to push their details into into a CRM.

Also, if this is possible and the conversion is triggered by a form mapped to a contact, will the contact already contain updates from the form?

Or, alternatively, am I going about this entirely the wrong way?!