The right upper bound conjecture for automatic complexity structure functions

Let $h_x(m)$ be the minimum number of states of a nondeterministic finite automaton accepting $x$ and at most $b^m$ strings of length $\lvert x\rvert$, let $g_x(m)=h_x(\lvert x\rvert-m)$, and let $G_n(m)=\sup_{\lvert x\rvert=n}g_x(m)$. The right upper bound conjecture. For every fixed integer $k$,

$$\lim_{n\to\infty}G_n(k)=k+1.$$

This conjecture predicts the asymptotic maximum of the converse structure function at each fixed parameter $k$; the supplied text gives no evidence that it has been resolved.