Generalized unprovability of Herbrand consistency for ${\rm IDelta}_0$

Let ${\rm IDelta}_0$ be the bounded-induction theory of arithmetic, let ${\rm \Omega}_n$ denote the corresponding family of axioms, and let ${\rm HCon}^*({\rm IDelta}_0)$ denote the paper's notion of Herbrand consistency for ${\rm IDelta}_0$. Consider the union of the theories ${\rm IDelta}_0+{\rm \Omega}_n$ over all $n$. Generalized Herbrand-consistency unprovability conjecture.

$$\bigcup_n({\rm IDelta}_0+{\rm \Omega}_n)\not\vdash {\rm HCon}^*({\rm IDelta}_0).$$

The conjecture proposes extending Kołodziejczyk's unprovability result using the paper's coding techniques and definitions of Herbrand consistency. Its status is not determined by the supplied source context.