I waited for 25.11 to get the permission check logic in place. Now that's resolved, there appears to be a new issue, but based on the error alone, I can't be 100% sure on what it is, although I'm betting it's tied to permissions.
The instance shows healthy, but it fails to run due to a connection string issue. See error:
Updating certificates in /etc/ssl/certs...
142 added, 0 removed; done.
Running hooks in /etc/ca-certificates/update.d...
done.
info: Microsoft.Hosting.Lifetime[0]
      Application started. Press Ctrl+C to shut down.
info: Microsoft.Hosting.Lifetime[0]
      Hosting environment: Production
info: Microsoft.Hosting.Lifetime[0]
      Content root path: /usr/local/proget
Initializing embedded database...
fail: Microsoft.Extensions.Hosting.Internal.Host[11]
      Hosting failed to start
      System.InvalidOperationException: The ConnectionString property has not been initialized.
         at Npgsql.ThrowHelper.ThrowInvalidOperationException(String message)
         at Npgsql.NpgsqlConnection.Open(Boolean async, CancellationToken cancellationToken)
         at Npgsql.NpgsqlConnection.Open()
         at Inedo.ProGet.Data.PostgresDatabaseContext.CreateConnection() in C:\Users\builds\AppData\Local\Temp\InedoAgent\BuildMaster\192.168.44.60\Temp\_E597550\Src\src\ProGet\Data\PostgresDatabaseContext.cs:line 58
         at Inedo.ProGet.Data.VirtualDatabaseContext.PostgresContext.CreateConnection() in C:\Users\builds\AppData\Local\Temp\InedoAgent\BuildMaster\192.168.44.60\Temp\_E597550\Src\src\ProGet\Data\VirtualDatabaseContext.cs:line 49
         at Inedo.ProGet.Data.VirtualDatabaseContext.CreateConnection() in C:\Users\builds\AppData\Local\Temp\InedoAgent\BuildMaster\192.168.44.60\Temp\_E597550\Src\src\ProGet\Data\VirtualDatabaseContext.cs:line 24
         at Inedo.Data.DatabaseContext.ExecuteInternal(String storedProcName, GenericDbParameter[] parameters, DatabaseCommandReturnType returnType)
         at Inedo.Data.DatabaseContext.ExecuteNonQuery(String storedProcName, GenericDbParameter[] parameters)
         at Inedo.Data.DatabaseContext.ExecuteScalar[TResult](String storedProcName, GenericDbParameter[] parameters, Int32 outParameterIndex)
         at Inedo.ProGet.Data.DB.Context.CheckSqlServerDbo(Nullable`1 IsOwner_Indicator) in C:\Users\builds\AppData\Local\Temp\InedoAgent\BuildMaster\192.168.44.60\Temp\_E597550\Src\src\ProGet\obj\Release\net8.0\linux-x64\InedoLib.Analyzers\InedoLib.Analyzers.DatabaseContextGenerator\DB.g.cs:line 2191
         at Inedo.ProGet.Data.DB.CheckSqlServerDbo(Nullable`1 IsOwner_Indicator) in C:\Users\builds\AppData\Local\Temp\InedoAgent\BuildMaster\192.168.44.60\Temp\_E597550\Src\src\ProGet\obj\Release\net8.0\linux-x64\InedoLib.Analyzers\InedoLib.Analyzers.DatabaseContextGenerator\DB.g.cs:line 183
         at Inedo.ProGet.Data.DatabaseMan.CheckConnection(Boolean asyncWithRetryMode, CancellationToken cancellationToken) in C:\Users\builds\AppData\Local\Temp\InedoAgent\BuildMaster\192.168.44.60\Temp\_E597550\Src\src\ProGet\Data\DatabaseMan.cs:line 62
         at Inedo.ProGet.Service.ProGetService.OnStartAsync(CancellationToken cancellationToken) in C:\Users\builds\AppData\Local\Temp\InedoAgent\BuildMaster\192.168.44.60\Temp\_E597550\Src\src\ProGet\Service\ProGetService.cs:line 40
         at Inedo.ProGet.Service.ProGetService.ExecuteAsync(CancellationToken stoppingToken) in C:\Users\builds\AppData\Local\Temp\InedoAgent\BuildMaster\192.168.44.60\Temp\_E597550\Src\src\ProGet\Service\ProGetService.cs:line 26
         at Microsoft.Extensions.Hosting.Internal.Host.<StartAsync>b__15_1(IHostedService service, CancellationToken token)
         at Microsoft.Extensions.Hosting.Internal.Host.ForeachService[T](IEnumerable`1 services, CancellationToken token, Boolean concurrent, Boolean abortOnFirstException, List`1 exceptions, Func`3 operation)
And config (note, I added the SSL Certificate directory as a mounted volume to avoid a different permission problem when running the certificate updates).
spec:
  replicas: 1
  selector:
    matchLabels:
      app: proget
  template:
    metadata:
      labels:
        app: proget
    spec:
      securityContext:
        runAsUser: 1024
        runAsGroup: 100
        runAsNonRoot: false
      containers:
        - name: proget
          image: proget.inedo.com/productimages/inedo/proget:latest
          ports:
            - containerPort: 8624
              name: http
          volumeMounts:
            - name: proget-data
              mountPath: /var/proget/database
            - name: proget-package
              mountPath: /var/proget/packages
            - name: proget-backup
              mountPath: /var/proget/backups
            - name: proget-certs
              mountPath: /etc/ssl/certs
      volumes:
        - name: proget-data
          persistentVolumeClaim:
            claimName: proget-data
        - name: proget-package
          persistentVolumeClaim:
            claimName: proget-package
        - name: proget-backup
          persistentVolumeClaim:
            claimName: proget-backup
        - name: proget-certs
          persistentVolumeClaim:
            claimName: proget-certs