I focused on statement F because it seems the least convoluted.

Suppose D-F are all false.

Because F is false, A is true. If A is true, then B is true.

Thus, if D-F are all false, then A and B are both true.

We just derived statement C without any assumptions*, thus C is absolutely true.

*The assumption about D-F all false is part of the assertion in Statement C.

So C is true in the same sense that "Shroedinger's cat is alive if the cyanide didn't trigger" is true. This is independent of whether or not the cyanide triggered or if the cat is dead or alive.

Suppose F is false.

Because F is false, A is true. If A is true, B is true.

We also know C is true from the prior deduction.

The possible solutions at this point are A,B,C,?D,?E,~F.

If E is true, D has to be true because F is false. However that will contradict D.

If E is false, D can be true or false and stay consistent with itself.

However B will be inconsistent if D is true, so D must be false.

Thus, given F is false, the solution would be A,B,C,~D,~E,~F.

Suppose F is true.

Because F is true, A is false.

We know C is true from the prior deduction.

Because C and F are both true, B must be false or it will contradict itself.

The possible solutions at this point are ~A,~B,C,?D,?E,F.

If D is true, E must be true to keep D consistent.

However E will be inconsistent if D and F are both true, so D must be false.

If D is false and F is true, E must be true.

Thus, given F is true, the solution would be ~A,~B,C,~D,E,F.

F has to be either true or false (otherwise this problem would have no correct answer).

We just considered both cases, F is false and F is true. In both cases that led to a solution with three truths. Thus, the answer is 3.