If there's a nuclear conflict involving >1,000 detonations, will humanity be extinct within 50 years?

Related Questions on Metaculus:

• If there's a nuclear conflict involving >1000 detonations, will humanity's population be <400 million 50 years later?

• Nuclear GC to cause (near) extinction

• Would we recover if population falls <400m?

• Will humans go extinct by 2100?

• Extinction if population falls <400 million?

As stated on another question about extinction:

"N.B. Even though it is obviously the case that if human extinction occurs Metaculus points won't be very valuable anymore and that it will be practically impossible to check for true human extinction (zero humans left), I would like to ask people not to let this fact influence their prediction and to predict in good faith."