Can you cast Speak With Dead on dead undead?

The spell Speak With Dead has the following descriptive text:

You grant the semblance of life and intelligence to a corpse of your choice within range, allowing it to answer the questions you pose. The corpse must still have a mouth and can’t be undead. The spell fails if the corpse was the target of this spell within the last 10 days.

Until the spell ends, you can ask the corpse up to five questions. The corpse knows only what it knew in life, including the languages it knew. Answers are usually brief, cryptic, or repetitive, and the corpse is under no compulsion to offer a truthful answer if you are hostile to it or it recognizes you as an enemy. This spell doesn’t return the creature’s soul to its body, only its animating spirit. Thus, the corpse can’t learn new information, doesn’t comprehend anything that has happened since it died, and can’t speculate about future events. (emphasis mine)

So let’s say my party is tracking someone with knowledge they need, who unbeknownst to them was betrayed by the big bad, killed and turned into a zombie. Obviously in zombie form he can’t be targeted by the spell, because he’s undead.

Once they reduce him to 0 hp (assuming there’s enough of the corpse left to talk with), is he still undead, and therefore unable to be the target of the Speak With Dead spell? Or is he just an ordinary corpse at that point, and this a valid target for the spell? Are dead undead still undead, or are they just dead?