fix(studio): build worker URLs on the project domain and correct the region

Workers answer at <project>/workers/v1/<name>, not a shared workers.supabase.co
host, and the alpha region is us-west-2.

apps/studio/components/interfaces/Workers/Workers.constants.ts

@@ -1,12 +1,21 @@
 import type { Worker, WorkerBuildState } from './Workers.types'
 
 // Locked to one region at alpha, so nothing in the UI lets you change it.
-export const WORKERS_REGION = 'us-west-1'
+export const WORKERS_REGION = 'us-west-2'
 export const WORKERS_REGION_LABEL = 'US West (Oregon)'
 export const WORKERS_REGION_SHORT = 'US West'
 
-// Private workers share this URL shape; access is enforced by the gateway, not the path.
-export const workerGatewayUrl = (name: string) => `https://workers.supabase.co/v1/${name}`
+// Workers answer on the project's own domain, alongside /functions/v1, and the api-gateway keeps
+// the route dark for projects outside the alpha allow-list.
+export const workerUrl = ({
+  endpoint,
+  protocol = 'https',
+  name,
+}: {
+  endpoint: string | undefined
+  protocol?: string
+  name: string
+}) => (endpoint ? `${protocol}://${endpoint}/workers/v1/${name}` : undefined)
 
 export const LISTENING_PORT = 8080
 

apps/studio/components/interfaces/Workers/Workers.utils.test.ts

@@ -1,5 +1,6 @@
 import { describe, expect, it } from 'vitest'
 
+import { workerUrl } from './Workers.constants'
 import type { Worker } from './Workers.types'
 import {
   filterWorkers,
@@ -137,3 +138,21 @@ describe('isWorkersAccessDenied', () => {
     expect(isWorkersAccessDenied(null)).toBe(false)
   })
 })
+
+describe('workerUrl', () => {
+  it('answers on the project domain alongside /functions/v1', () => {
+    expect(workerUrl({ endpoint: 'abcdefgh.supabase.co', name: 'embed' })).toBe(
+      'https://abcdefgh.supabase.co/workers/v1/embed'
+    )
+  })
+
+  it('honours a non-https protocol', () => {
+    expect(workerUrl({ endpoint: 'localhost:8000', protocol: 'http', name: 'embed' })).toBe(
+      'http://localhost:8000/workers/v1/embed'
+    )
+  })
+
+  it('has no url until the project settings resolve', () => {
+    expect(workerUrl({ endpoint: undefined, name: 'embed' })).toBeUndefined()
+  })
+})