[WebToolsE2E][Aspire] After successfully publishing an Aspire project, the Container app Start time doesn't match the current time after restarting the project in the 'Aspire Dashboard'

**INSTALL STEPS**
1. Clean machine: Win11 x64 23h2 ENU
2. Install SDK 8.0.400
3. Install Aspire 8.1.0
4. Install latest released azd v1.9.6
5. Install git v2.46.0

**REPRO STEPS**
1. Open CMD, run following command to create an Aspire Starter App project and deploy it
```
dotnet new aspire-starter -o AspireStarterApp
cd AspireStarterApp/AspireStarterApp.AppHost
azd init
azd up
```
2. After successful deployment, open the deployed "Aspire Dashboard" URL
3. In the Aspire dashboard resources, remember the Container app Start time, then Restart 'apiservice' under Commands

**Note:**
1. After refreshing the page, the Container app Start time matches the current time

**ACTUAL**
During the restart process, the Start time matches the current time. After the restart is complete, the Container app Start time doesn't match the current time

**EXPECTED**
The Container app Start time matches the current time after the project restart is complete