diff --git a/modules/mesh-delivery-assigner/internal/assign/wants.go b/modules/mesh-delivery-assigner/internal/assign/wants.go
index 2903a82..bd736c6 100644
--- a/modules/mesh-delivery-assigner/internal/assign/wants.go
+++ b/modules/mesh-delivery-assigner/internal/assign/wants.go
@@ -316,9 +316,11 @@ func Wants(m Module, n Nodes, wants map[string]delivery.Want, lastHealthy map[st
 				ok, target = true, have.FP
 			}
 		}
-		if ok && target == m.FP && len(out.NodeFailed) > 0 && !(had && !before.Removed && before.FP == m.FP) {
-			// While a node fails F alone, F reaches no further node: a node whose want does not name F yet gets no new want
-			// naming it, and wants already written stay (ADR 0557 §2).
+		if ok && target == m.FP && len(out.NodeFailed) > 0 && !(had && !before.Removed && before.FP == m.FP) &&
+			n.Haves[node][m.Name].FP != m.FP {
+			// While a node fails F alone, F reaches no further node: a node that neither wants nor runs F gets no new want
+			// naming it, and wants already written stay (ADR 0557 §2). A node that runs F with no live want gets its want at F, or
+			// it would be on neither list and its node-engine would give the module up (issue 1945).
 			ok, rule, words = false, delivery.RuleNotHeld, FailedAloneWords(m.Name, m.FP, slices.Sorted(maps.Keys(out.NodeFailed)))
 		}
 		switch {
diff --git a/modules/mesh-delivery-assigner/internal/assign/wants_test.go b/modules/mesh-delivery-assigner/internal/assign/wants_test.go
index 3e6985b..ff629e2 100644
--- a/modules/mesh-delivery-assigner/internal/assign/wants_test.go
+++ b/modules/mesh-delivery-assigner/internal/assign/wants_test.go
@@ -183,6 +183,13 @@ func TestAFailedHealthHoldsAndSetsBack(t *testing.T) {
 	if _, written := out.Wants["g14"]; written {
 		t.Errorf("g14's want at F was rewritten: %+v", out.Wants["g14"])
 	}
+	// A node that runs F healthy with no want, while g14 fails it alone, gets its want at F and is on no neither list (issue 1945).
+	n.Haves["novox"]["mail"] = delivery.Have{FP: fNew, Phase: delivery.PhaseRunning, Health: delivery.HealthOK}
+	delete(before, "novox")
+	out = Wants(three, n, before, map[string]string{"ace": fOld, "g14": fOld, "novox": fOld})
+	if w := out.Wants["novox"]; w.FP != fNew || len(out.Neither) != 0 {
+		t.Errorf("novox running F with no want: want %+v, neither %+v; want its want at F and no neither", w, out.Neither)
+	}
 	n.Haves["ace"]["mail"], n.Haves["g14"]["mail"] = failedHave, n.Haves["ace"]["mail"]
 
 	// ADR 0557 §2: a witness that is offline keeps the have it last wrote and witnesses nothing (D1); F failing on a second node is