Is it possible to have 2 ASP.NET Core sites in one Web App?

Is it possible to have 2 ASP.NET Core Web API apps inside the one App Service?

I understand that it would be better to have the 2 sites in separate App Services, but our use case is a bit unique.

We're trying to add the 2nd site for the purposes of using the applicationInitialization feature to introduce a delay between our app starting and the instance being given to the App Service load balancer. I could go into a lot more detail on this, but what I desire for now is to bolt-on an additional web site that the applicationInitialization can call to delay the startup process.

I've tried:

Folder structure: 
  /mySite
  /mySite2
  web.config

Folder structure: 
  /mySite2
  mySite.dll
  web.config

With respective web.Configs looking like:

<?xml version="1.0" encoding="utf-8"?>
<configuration>
  <location path="mySite" inheritInChildApplications="false">
    <system.webServer>
      <handlers>
        <add name="aspNetCore" path="*" verb="*" modules="AspNetCoreModuleV2" resourceType="Unspecified" />
      </handlers>
      <aspNetCore processPath="dotnet" arguments=".\mysite\mySite.dll" stdoutLogEnabled="false" stdoutLogFile=".\logs\stdout" hostingModel="OutOfProcess" />
    </system.webServer>
  </location>
  <location path="mySite2" inheritInChildApplications="false">
    <system.webServer>
      <handlers>
        <add name="aspNetCore" path="*" verb="*" modules="AspNetCoreModuleV2" resourceType="Unspecified" />
      </handlers>
      <aspNetCore processPath="dotnet" arguments=".\mySite2\mySite2.dll" stdoutLogEnabled="false" stdoutLogFile=".\logs\stdout" hostingModel="OutOfProcess" />
    </system.webServer>
  </location>
</configuration>

and

<?xml version="1.0" encoding="utf-8"?>
<configuration>
  <location path="." inheritInChildApplications="false">
    <system.webServer>
      <handlers>
        <add name="aspNetCore" path="*" verb="*" modules="AspNetCoreModuleV2" resourceType="Unspecified" />
      </handlers>
      <aspNetCore processPath="dotnet" arguments=".\mySite.dll" stdoutLogEnabled="false" stdoutLogFile=".\logs\stdout" hostingModel="InProcess" />
    </system.webServer>
  </location>
  <location path="mySite2" inheritInChildApplications="false">
    <system.webServer>
      <handlers>
        <add name="aspNetCore" path="*" verb="*" modules="AspNetCoreModuleV2" resourceType="Unspecified" />
      </handlers>
      <aspNetCore processPath="dotnet" arguments=".\mySite2\mySite2.dll" stdoutLogEnabled="false" stdoutLogFile=".\logs\stdout" hostingModel="OutOfProcess" />
    </system.webServer>
  </location>
</configuration>