Are USB 2.0 hubs required to disable downstream VBUS when upstream VBUS is removed
External power in case of hubs refers to external power adapter (or even a battery). If this is the only power to the hub, and it is a solo source of Vcc for hub controller IC, then it is obvious that downstream ports will have no power. The hub will be simply dead. 

The confusion comes in cases when a hub has dual functionality and can derive the power either from external source, or from upstream VBUS, aka "power switching" hub. In accord with specifications, there should be some switch which gracefully replaces external power with VBUS power. When this happens, the hub must turn all downstream ports off. Also in this case all hubs must change the content of their descriptors from being "self-powered" to "bus-powered". Bus-powered hubs have power restrictions, so the host should know this, and apply corresponding power policy during re-enumeration process. 

The question, however, is challenging. When VBUS is removed, the hub, as an any USB device, becomes "detached". USB specifications do not define behavior of devices in detached state, see Section 9.1.1.1. So this state is open to interpretations and up to designers.

One interpretation is that when a hub is not connected to bus, it is not attached, and and therefore cannot remain configured. As Section 11.13 says, 


  If a hub implements power switching, no power is provided to the
  downstream facing ports while the hub is not configured.


So, not configured = downstream VBUS is OFF. AFAIK, Microchip follows this interpretation when upstream VBUS is disconnected. There are other hubs with ports that can maintain VBUS, but they are called "charging ports" and are formally outside the USB framework. To support the logic of bus/self-powered switch, these chips have a special pin called LOCAL_PWR.

Hubs that don't use port power switching have VBUS on downstream ports all the time. I believe this is a violation of USB specifications, that's why you can't find the USB-IF certification logo on any cheap tiny hubs.