Next.js + Supabase: user looks signed out after refreshing the page

// middleware.ts
import { NextResponse, type NextRequest } from "next/server";
import { createServerClient } from "@supabase/ssr";

export async function middleware(request: NextRequest) {
  let response = NextResponse.next({ request });
  const supabase = createServerClient(process.env.NEXT_PUBLIC_SUPABASE_URL!, process.env.NEXT_PUBLIC_SUPABASE_ANON_KEY!, {
    cookies: {
      getAll: () => request.cookies.getAll(),
      setAll: (list) => {
        list.forEach(({ name, value }) => request.cookies.set(name, value));
        response = NextResponse.next({ request });
        list.forEach(({ name, value, options }) => response.cookies.set(name, value, options));
      },
    },
  });
  await supabase.auth.getClaims(); // refreshes and saves the session
  return response;
}