Conf_path is deprecated but what is its replacement?
One of two change records for conf_path() is Bootstrap refactored to use the kernel. This states that


  Note: In D8, instead of using conf_path(), \Drupal\Core\DrupalKernel::getSitePath() instead. If the kernel is unavailable (such as above, where the code shows the booting of the D8 kernel) or the site path needs to be recalculated then Drupal\Core\DrupalKernel::findSitePath() can be used


So if you have access to the DrupalKernel object, then calling getSitePath() on that object will work. This is probably not the case. 

However, there is a service for this called site.path that can be injected into a class that you need it. I think the change record could be updated to reference this factory class.

Using the global Drupal object:

$conf_path = \Drupal::service('site.path');