Registry: How to get notified about a crashed process after a LiveView refresh?

I’m doing some experimentation, and I ran into an issue with my Registry setup.

I have an event where I start some doomed tasks:

def handle_event("start_processes", _, socket) do
  for _ <- 1..5 do
    name = :rand.bytes(5) |> Base.url_encode64(padding: false)

    task =
      TaskSupervisor
      |> Task.Supervisor.async_nolink(fn ->
        Process.sleep(to_timeout(second: :rand.uniform(9)))
        raise "Crash!"
      end)

    Registry.register(ProcessesRegistry, name, task.pid)

    task
  end
  processes = Registry.select(ProcessesRegistry, registry_select_all())
  {:noreply, assign(socket, processes: processes)}
end

I’d like to update my LiveView about the status of these tasks:

<div>
  <p>Current process: {inspect(self())}</p>
  <p>Registry process: {inspect(Process.whereis(ProcessesRegistry))}</p>

  <.button phx-click="get_registered_processes">Refresh Processes</.button>
  <.button phx-click="start_processes">Start Processes</.button>

  <h2>Background Processes</h2>
  <%= for pid <- @processes do %>
    <div>PID: {inspect(pid)}</div>
  <% end %>
</div>

def handle_info({:DOWN, _ref, :process, pid, _reason}, socket) do
  IO.puts("handling DOWN message in #{inspect(socket.root_pid)}")

  [{key, {_, _}}] = Registry.select(ProcessesRegistry, [{{:_, :_, pid}, [], [:"$_"]}])
  Registry.unregister(ProcessesRegistry, key)

  processes = Registry.select(ProcessesRegistry, registry_select_all())
  {:noreply, assign(socket, processes: processes)}
end

But there is a very simple problem: what happens when the LiveView page is refreshed?

The PID of the LiveView is changed, which causes the :DOWN messages to be sent to the old LiveView process instead of the new one.

Is there a simple way to solve this problem?

Thanks