Application Insights does not log dependencies with a profiler attached

[VladimirAkopyan]

• List of NuGet packages and version that you are using:

1. ApplicationInsights.AspNetCore 2.22
2. NewRelic.Agent 10.23.0
3. Swashbuckle.AspNetCore

• Runtime version: .Net 8
• Hosting environment: Docker Desktop and Azure Kubernetes Service

Describe the bug

Application insights does not log some Dependency HTTP requests when NewRelic agent is installed and Profiler is attached.

Our company has been using Applicaiton Insights and NewRelic in parallel on .Net Core 2 and 3.1.
During out migration to .Net 8 we have discovered that some dependencies have got missing.
This happens to some HTTP requests, but not others - it appears to depend on the URL. For example HTTP GET request to google.com will always be logged, but HTTP GET to api.restful-api.dev/objects will not, for no obvious reason.

To Reproduce

I have created a project using ASP.Net 8 API template, with Docker Compose, Application Insights and NewRelic Agent.
You will need to set a New Relic License key and an Application Insights connection string to test.

Run the application, and open the swagger UI. Make the default GET request
Upon receiving the GET request, the .Net Application will make several API requests

var client = new HttpClient();
await client.GetAsync($"{baseURL}");
await client.PostAsJsonAsync($"{baseURL}posts", "{ \"phone\": \"+1 (975) 458-3685\"}");
await client.PutAsJsonAsync($"{baseURL}posts/1", "{ \"phone\": \"+1 (975) 458-3685\"}");
await client.PatchAsJsonAsync($"{baseURL}posts/1", "{ \"phone\": \"+1 (975) 458-3685\"}");
await client.DeleteAsync($"{baseURL}posts/1");
await client.GetAsync($"{baseURL}posts");
await client.GetAsync("https://google.com");
await client.GetAsync("https://api.restful-api.dev/objects");

Check Application Insights, and you will see that only the GET request to google is logged.
Other dependencies, like SQL and service bus, are logged successfully.

In my understanding, this Application Insights Package does not attach as a profiler and thus I am unclear why it is affected.
If New Relic Agent is removed, the issue dissapears.

[VladimirAkopyan]

The sample here demonstrates the error: https://github.com/VladimirAkopyan/NewRelic_AppInsights_Conflict