diff --git a/NEWS b/NEWS
index debc196c7..af4287184 100644
--- a/NEWS
+++ b/NEWS
@@ -1,6 +1,13 @@
 New in spot 2.9.1.dev  (not yet released)
 
-  Nothing yet.
+  Bugs fixed:
+
+  - ltlcross --csv=... --product=N with N>0 could output spurious
+    diagnosics claiming that words were rejected when evaluating
+    the automaton on state-space.
+
+  - spot::scc_info::determine_unknown_acceptance() now also update the
+    result of spot::scc_info::one_accepting_scc().
 
 New in spot 2.9.1 (2020-07-15)
 
diff --git a/bin/ltlcross.cc b/bin/ltlcross.cc
index fd8deb228..784f06b2f 100644
--- a/bin/ltlcross.cc
+++ b/bin/ltlcross.cc
@@ -1568,6 +1568,7 @@ namespace
                         sm = new
                           spot::scc_info(p,
                                          spot::scc_info_options::TRACK_STATES);
+                        sm->determine_unknown_acceptance();
                       }
                     catch (const std::bad_alloc&)
                       {
diff --git a/spot/twaalgos/sccinfo.cc b/spot/twaalgos/sccinfo.cc
index 024051bcb..74498c49f 100644
--- a/spot/twaalgos/sccinfo.cc
+++ b/spot/twaalgos/sccinfo.cc
@@ -472,9 +472,15 @@ namespace spot
                   "does not support alternating automata");
             auto& node = node_[s];
             if (check_scc_emptiness(s))
-              node.rejecting_ = true;
+              {
+                node.rejecting_ = true;
+              }
             else
-              node.accepting_ = true;
+              {
+                node.accepting_ = true;
+                if (one_acc_scc_ < 0)
+                  one_acc_scc_ = s;
+              }
             changed = true;
           }
       }
diff --git a/spot/twaalgos/sccinfo.hh b/spot/twaalgos/sccinfo.hh
index a76ca3612..0b70b29d5 100644
--- a/spot/twaalgos/sccinfo.hh
+++ b/spot/twaalgos/sccinfo.hh
@@ -535,7 +535,14 @@ namespace spot
       return node_.size();
     }
 
-    /// Return the number of one accepting SCC if any, -1 otherwise.
+    /// \brief Return the number of one accepting SCC if any, -1 otherwise.
+    ///
+    /// If an accepting SCC has been found, return its number.
+    /// Otherwise return -1.  Note that when the acceptance condition
+    /// contains Fin, -1 does not implies that all SCCs are rejecting:
+    /// it just means that no accepting SCC is known currently.  In
+    /// that case, you might want to call
+    /// determine_unknown_acceptance() first.
     int one_accepting_scc() const
     {
       return one_acc_scc_;
diff --git a/tests/core/ltlcross3.test b/tests/core/ltlcross3.test
index 430107239..652c9f88b 100755
--- a/tests/core/ltlcross3.test
+++ b/tests/core/ltlcross3.test
@@ -327,6 +327,11 @@ diff foo expected
 # This command used to crash.  Report from František Blahoudek.
 run 0 ltlcross --verbose --no-checks -f 'FGa' 'ltl2tgba'
 
+# In spot 2.9 and 2.9.1 the following command used to report a bug
+# that did not exist.  Issue #420.
+ltlcross --csv=/dev/null ltl2tgba 'ltl2tgba -D -G' --states=5 \
+         -f '(G(F(((a) & (X(X(a)))) | ((!(a)) & (X(X(!(a))))))))'
+
 # The CSV file should not talk about product if --products=0
 ltlcross --color --products=0 ltl2tgba -f GFa -f FGa --csv=out.csv
 grep product out.csv && exit 1