Failure of initial algebras for monic polynomials with reductions

Let a topos with natural number object be a topos equipped with a natural number object, and let a monic polynomial with reductions be as considered in the paper. Initial-algebra failure conjecture.

1. There is a topos with natural number object with a monic polynomial with reductions that does not have an initial algebra.
2. It is consistent with ${IZF}$ that there is a monic polynomial with reductions in the category of sets that does not have an initial algebra.

These are concrete forms of the claim that choice may be necessary for constructing initial algebras. The paper observes that WISC must fail in the proposed topos and rules out Boolean toposes and internal presheaf categories over Boolean toposes; the conjecture remains open.