Trace to Profile in aspnetcore nested Activities

The PyroscopeSpanProcessor works like a charm with aspnetcore, I see the pyroscope.profile.id in the root span and the linked profile includes the cpu/mem intensive call stack

        [HttpGet]
        public async Task<string> Get()
        {
            ... cpu/mem intensive work ...
        }

But it does not work when manually defining child Activities. With the following example I see the pyroscope.profile.id in the root span but the linked profile does not include the cpu/mem intensive call stack and there is no pyroscope.profile.id on the MyCustomActivity child span.

        [HttpGet]
        public async Task<string> Get()
        {
                using (var activity = ActivitySource.StartActivity("MyCustomActivity", ActivityKind.Internal, default(ActivityContext)))
                {
                     ... cpu/mem intensive work ...
                }
        }

Environment

.Net 8
opentelemetry.instrumentation.aspnetcore 1.8.1
pyroscope-dotnet v0.2.0-opentelemetry
pyroscope.0.8.14-glibc-x86_64

See: https://github.com/grafana/pyroscope-dotnet/blob/main/Pyroscope/Pyroscope.OpenTelemetry/PyroscopeSpanProcessor.cs#L12

Start by reading Pyroscope/Pyroscope.OpenTelemetry/PyroscopeSpanProcessor.cs at the linked location and reproduce the .NET 8 example with a manually created nested Activity. Compare the root and MyCustomActivity spans and the linked profile. Done means the child span receives pyroscope.profile.id and its CPU/memory-intensive call stack appears in the linked profile.