diff --git a/clients/drcachesim/scheduler/scheduler.cpp b/clients/drcachesim/scheduler/scheduler.cpp
index ffbb3004309..2ccd4b82a03 100644
--- a/clients/drcachesim/scheduler/scheduler.cpp
+++ b/clients/drcachesim/scheduler/scheduler.cpp
@@ -3274,14 +3274,11 @@ scheduler_tmpl_t<RecordType, ReaderType>::pick_next_input(output_ordinal_t outpu
                     } else {
                         auto lock =
                             std::unique_lock<mutex_dbg_owned>(*inputs_[prev_index].lock);
-                        // If we can't go back to the current input, we're EOF or idle.
-                        // TODO i#6959: We should go the EOF/idle route if
-                        // inputs_[prev_index].unscheduled as otherwise we're ignoring its
-                        // unscheduled transition: although if there are no other threads
-                        // at all (not just an empty queue) this turns into the
-                        // eof_or_idle() all-unscheduled scenario.  Once we have some kind
-                        // of early exit option we'll add the unscheduled check here.
-                        if (inputs_[prev_index].at_eof) {
+                        // If we can't go back to the current input because it's EOF
+                        // or unscheduled indefinitely (we already checked blocked_time
+                        // above: it's 0 here), this output is either idle or EOF.
+                        if (inputs_[prev_index].at_eof ||
+                            inputs_[prev_index].unscheduled) {
                             lock.unlock();
                             sched_type_t::stream_status_t status =
                                 eof_or_idle(output, prev_index);
diff --git a/clients/drcachesim/tests/scheduler_unit_tests.cpp b/clients/drcachesim/tests/scheduler_unit_tests.cpp
index b151c1ff9b9..f608cb48585 100644
--- a/clients/drcachesim/tests/scheduler_unit_tests.cpp
+++ b/clients/drcachesim/tests/scheduler_unit_tests.cpp
@@ -5222,6 +5222,62 @@ test_unscheduled_small_timeout()
     }
 }
 
+static void
+test_unscheduled_no_alternative()
+{
+    // Test that an unscheduled 0-timeout input is not incorrectly executed if
+    // there is nothing else to run (i#6959).
+    std::cerr << "\n----------------\nTesting unscheduled no alternative (i#6959)\n";
+    static constexpr int NUM_OUTPUTS = 1;
+    static constexpr uint64_t REBALANCE_PERIOD_US = 50;
+    static constexpr memref_tid_t TID_A = 100;
+    std::vector<trace_entry_t> refs_A = {
+        make_thread(TID_A),
+        make_pid(1),
+        make_version(TRACE_ENTRY_VERSION),
+        make_timestamp(1001),
+        make_marker(TRACE_MARKER_TYPE_CPU_ID, 0),
+        make_instr(/*pc=*/101),
+        make_timestamp(1002),
+        make_marker(TRACE_MARKER_TYPE_CPU_ID, 0),
+        make_marker(TRACE_MARKER_TYPE_SYSCALL, 999),
+        make_marker(TRACE_MARKER_TYPE_MAYBE_BLOCKING_SYSCALL, 0),
+        // No timeout means infinite (until the fallback kicks in).
+        make_marker(TRACE_MARKER_TYPE_SYSCALL_UNSCHEDULE, 0),
+        make_timestamp(2002),
+        make_instr(/*pc=*/102),
+        make_exit(TID_A),
+    };
+    {
+        std::vector<scheduler_t::input_reader_t> readers;
+        readers.emplace_back(std::unique_ptr<mock_reader_t>(new mock_reader_t(refs_A)),
+                             std::unique_ptr<mock_reader_t>(new mock_reader_t()), TID_A);
+        static const char *const CORE0_SCHED_STRING =
+            "...A......__________________________________________________A.";
+
+        std::vector<scheduler_t::input_workload_t> sched_inputs;
+        sched_inputs.emplace_back(std::move(readers));
+        scheduler_t::scheduler_options_t sched_ops(scheduler_t::MAP_TO_ANY_OUTPUT,
+                                                   scheduler_t::DEPENDENCY_TIMESTAMPS,
+                                                   scheduler_t::SCHEDULER_DEFAULTS,
+                                                   /*verbosity=*/3);
+        // We use our mock's time==instruction count for a deterministic result.
+        sched_ops.quantum_unit = scheduler_t::QUANTUM_TIME;
+        sched_ops.time_units_per_us = 1.;
+        sched_ops.rebalance_period_us = REBALANCE_PERIOD_US;
+        scheduler_t scheduler;
+        if (scheduler.init(sched_inputs, NUM_OUTPUTS, std::move(sched_ops)) !=
+            scheduler_t::STATUS_SUCCESS)
+            assert(false);
+        std::vector<std::string> sched_as_string =
+            run_lockstep_simulation(scheduler, NUM_OUTPUTS, TID_A, /*send_time=*/true);
+        for (int i = 0; i < NUM_OUTPUTS; i++) {
+            std::cerr << "cpu #" << i << " schedule: " << sched_as_string[i] << "\n";
+        }
+        assert(sched_as_string[0] == CORE0_SCHED_STRING);
+    }
+}
+
 static void
 test_unscheduled()
 {
@@ -5230,6 +5286,7 @@ test_unscheduled()
     test_unscheduled_initially();
     test_unscheduled_initially_roi();
     test_unscheduled_small_timeout();
+    test_unscheduled_no_alternative();
 }
 
 static void