Two kinds of work, and a switch to say which

The sidebar rendered an agent chat and an ordinary one identically, in one
list, so hours of machine work sat among a morning's questions. A switch
below the pinned models now shows one kind at a time, stored on the account
so it follows the reader to another browser.

Three things it does that are not the obvious version:

The switch is inside the fragment it swaps. Targeting only the tree would
leave the two buttons showing the side you had just left -- the request
works and the interface says otherwise, which is the failure this codebase
keeps cataloguing.

A folder can be emptied by the filter, or have been empty all along, and
only the first is a reason to hide it. `shown_in` is that line: a folder
somebody made a moment ago and has not filled yet stays on both sides, or
it can never be found again, let alone filed into.

With agent chats switched off there is no switch, and the sidebar goes back
to showing everything rather than to one side of a fork nobody can move.
An administrator turning the feature off would otherwise strand whoever
last left the switch on Agents in an empty sidebar with no way out.

The control reuses the composer's `.segmented`, which is the same choice in
a different place, and the verb goes on the input rather than the wrapper.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>

src/lembas/web/templates/chat/_composer.html

@@ -275,15 +275,22 @@
         #}
         {% if not chat and agent_profiles %}
         <div class="composer__context" data-agent-picker>
-          <input type="hidden" name="kind" value="chat" id="chat-kind">
+          {# Seeded from `?kind=`, which is how the sidebar's Agent side opens
+             this screen already on the right fork. `ui.js` reads the checked
+             radio when it wires the picker, so the hidden field and the
+             revealed connection follow from this and nothing else. #}
+          <input type="hidden" name="kind" value="{{ starting_kind | default('chat') }}"
+                 id="chat-kind">
 
           <div class="segmented" role="group" aria-label="Kind of chat">
             <label class="segmented__option">
-              <input type="radio" name="kind_choice" value="chat" checked>
+              <input type="radio" name="kind_choice" value="chat"
+                     {{ '' if starting_kind == 'agent' else 'checked' }}>
               <span>{{ icon("chat", "icon--sm") }} Chat</span>
             </label>
             <label class="segmented__option">
-              <input type="radio" name="kind_choice" value="agent">
+              <input type="radio" name="kind_choice" value="agent"
+                     {{ 'checked' if starting_kind == 'agent' }}>
               <span>{{ icon("bolt", "icon--sm") }} Agent</span>
             </label>
           </div>