Bounded deficiency and unbounded price conjecture for nondeterministic automatic complexity

Let $A_N$ denote nondeterministic automatic complexity. For a random infinite binary sequence, let $D_n$ be the nondeterministic automatic complexity deficiency of its length-$n$ prefix, and let $V_n$ be the associated price. Bounded deficiency and unbounded price conjecture.

$$\mathbb P(\sup_n D_n<\infty)=1,\qquad \sup_n V_n=\infty.$$

This conjecture proposes that finite complexity deficiency is almost surely attained along all prefixes, even though the associated price is unbounded. It is presented as an open conjecture in the paper; the surrounding discussion motivates infinite price paths through this distinction.