AKS - combine addon HTTP application routing with nginx rewrite rule?

I'm using the Azure AKS addon for HTTP application routing as described here. I deployed it using Terraform and it generally works:

resource "kubernetes_ingress" "ingress" {
  metadata {
    name      = "nurse-ingress"
    namespace = kubernetes_namespace.nurse.metadata[0].name
    annotations = {
      "kubernetes.io/ingress.class" = "addon-http-application-routing"
      "nginx.ingress.kubernetes.io/rewrite-target" = "/$1"
    }
  }

  wait_for_load_balancer = true

  spec {
    backend {
      service_name = "nurse-service"
      service_port = 80
    }

    rule {
      host = "nurseapp.${azurerm_kubernetes_cluster.main.addon_profile[0].http_application_routing[0].http_application_routing_zone_name}"
      http {
        path {
          backend {
            service_name = kubernetes_service.app.metadata[0].name
            service_port = 80
          }

          path = "/app/(.*)"
        }

        path {
          backend {
            service_name = kubernetes_service.nurse.metadata[0].name
            service_port = 80
          }

          path = "/nurse/(.*)"
        }
      }
    }
  }
}

However, it only works on the default backend (i.e. path=/). When I call the URL on /nurse or /app it does not work since the rewrite-target /$1 does not seem to be taken into account. I will just get a 404 - since the nurse-service itself does expect calls on /foo and not on /nurse/foo

Should this be possible to configure to begin with and if so, any idea where my mistake is?