We ware struggling getting the HTTP API cron task to run. We found that (at the time of writing) the official documentation doesn't work for our version: 1.15.8. After many tries we found the correct way to call the cron API:
curl -d "{}" -H "X-API-Key: API_KEY_HERE " https://YOUR_DOMAIN/api/tasks/cron
In short, it needs to be a POST request, that has the X-API-Key header. Obviously you'll need to generate the API key for the IP you're executing the request from.
I hope I saved you some time.