Using router.reload with Laravel, Inertia and React

I am trying to lazy load part of the data in the controller. I am trying the below:

return Inertia::render('TeamMembers', [
                'test1=> "123",
                'testLazy' => Inertia::lazy(fn () =>
                    User::where('company_id', '=', $id)->get()
                )
            ],
        );

I expect, that the testLazy props will not be loaded on the first load, and then I can use useEffect to re-request only the testLazy, as follows:

router.reload({only: ['testLazy']});

But the problem is, it doesn't work and results in a 409. The only headers I see in the request are:

x-inertia: true
x-inertia-partial-data: testLazy
x-requested-with: XMLHttpRequest

Judging by the documentation https://inertiajs.com/the-protocol and the code https://github.com/inertiajs/inertia/blob/5c9c2707dc5f2f1f36d8eff99d731b5e42dab021/packages/core/src/router.ts#L355-L356 there should be additional headers. If I try to pass the headers manually, they still don't show up.

        const inertiaPage = usePage();

            <button onClick={() => router.reload(
                {
                    only: ['testLazy'],
                    headers: {
                        "x-inertia-partial-component": inertiaPage.component, // doesn't show up
                        "x-test": "x-test2", // shows up
                        "version": inertiaPage.version || "not-found", // shows up
                        "component": inertiaPage.component // shows up
                    }
                }
            )}>Reload test</button>

Using Laravel 9, @inertiajs/react 1.0.0, laravel-mix 6.0.49 (tried building with Vite, no difference). Any ideas?