diff --git a/packages/api-e2e/test/helpers/server.ts b/packages/api-e2e/test/helpers/server.ts
index 204f74107..230d92df3 100644
--- a/packages/api-e2e/test/helpers/server.ts
+++ b/packages/api-e2e/test/helpers/server.ts
@@ -49,7 +49,9 @@ export interface FixtureServer {
  * - `ws /ws` — a WebSocket echo endpoint: text and binary messages are sent
  *   back as-is, and {@link WEBSOCKET_CLOSE_REQUEST} makes the server close the
  *   connection with {@link WEBSOCKET_CLOSE_CODE}. On `/ws/headers` the server
- *   first sends the upgrade request's headers as a JSON text message.
+ *   answers the first message with the upgrade request's headers as a JSON
+ *   text message instead (not sent on connect, where it could arrive before
+ *   the client attached a listener).
  */
 export function startFixtureServer(): Promise<FixtureServer> {
   const server = http.createServer((req, res) => {
@@ -111,11 +113,12 @@ export function startFixtureServer(): Promise<FixtureServer> {
       return
     }
     wss.handleUpgrade(req, socket, head, (ws) => {
-      if (pathname === '/ws/headers') {
-        ws.send(JSON.stringify(req.headers))
-      }
+      let sendHeaders = pathname === '/ws/headers'
       ws.on('message', (data, isBinary) => {
-        if (
+        if (sendHeaders) {
+          sendHeaders = false
+          ws.send(JSON.stringify(req.headers))
+        } else if (
           !isBinary
           && Buffer.isBuffer(data)
           && data.toString('utf8') === WEBSOCKET_CLOSE_REQUEST
diff --git a/packages/api-e2e/test/specs/opener.spec.ts b/packages/api-e2e/test/specs/opener.spec.ts
index d5e1b733c..6e2265eaa 100644
--- a/packages/api-e2e/test/specs/opener.spec.ts
+++ b/packages/api-e2e/test/specs/opener.spec.ts
@@ -41,6 +41,7 @@ describePlugin('opener', () => {
         await api.path.join(await api.path.appDataDir(), 'does-not-exist-e2e')
       )
     )
-    expect(message).toMatch(/os error 2/)
+    // the `std::io` NotFound message on Unix, the shell path lookup's on Windows
+    expect(message).toMatch(/os error 2|path doesn't exist/)
   })
 })
diff --git a/packages/api-e2e/test/specs/positioner.spec.ts b/packages/api-e2e/test/specs/positioner.spec.ts
index f2cfadf9f..d0e7986d3 100644
--- a/packages/api-e2e/test/specs/positioner.spec.ts
+++ b/packages/api-e2e/test/specs/positioner.spec.ts
@@ -101,13 +101,21 @@ describePlugin('positioner', { desktopOnly: true }, () => {
   )
 
   itWm('tray positions are relative to the reported tray rect', async () => {
-    // a tray "icon" in the middle of the window's monitor, far from any edge,
-    // so neither the OS-specific flips nor the constraint kick in
+    // The example's 1000x800 window barely fits the CI screens (1024x768 on
+    // Windows, 1280x1024 on Xvfb), so wherever the tray "icon" is it sticks out
+    // of the monitor and `moveWindowConstrained` clamps it back on both axes.
+    // The window is shrunk and the tray placed near the monitor's top left
+    // corner, far enough from the edges that both positions keep the window on
+    // screen and neither the OS-specific flips nor the constraint kick in.
     const tray = await tauri(async (api) => {
+      const win = api.window.getCurrentWindow()
+      await win.setSize(new api.dpi.LogicalSize(600, 400))
+      await new Promise((resolve) => setTimeout(resolve, 500))
       const monitor = await api.window.currentMonitor()
+      const size = await win.outerSize()
       const rect = {
-        x: monitor!.position.x + Math.trunc(monitor!.size.width / 2),
-        y: monitor!.position.y + Math.trunc(monitor!.size.height / 2),
+        x: monitor!.position.x + Math.trunc(size.width / 2) + 50,
+        y: monitor!.position.y + 50,
         width: 20,
         height: 20
       }
diff --git a/packages/api-e2e/test/specs/websocket.spec.ts b/packages/api-e2e/test/specs/websocket.spec.ts
index cb01f55a7..474ad4ee0 100644
--- a/packages/api-e2e/test/specs/websocket.spec.ts
+++ b/packages/api-e2e/test/specs/websocket.spec.ts
@@ -73,6 +73,7 @@ describePlugin('websocket', () => {
       const message = await new Promise<Message>((resolve, reject) => {
         setTimeout(() => reject(new Error('headers not received')), 5000)
         ws.addListener(resolve)
+        ws.send('headers').catch(reject)
       })
       await ws.disconnect()
       return JSON.parse(message.data as string) as Record<string, string>