Asymptotic gap between monotone and non-monotone formula complexity

Let $a_0=2-1$ and, for $n\geq 0$, let

$$a_{n+1}=2^{1+a_n}-1.$$

Consider formula encodings whose gates are restricted to \{+,\times,\text{^}\} and whose inputs are restricted to $\{-1,1\}$. The asymptotic-gap conjecture. The gap between the monotone and non-monotone complexity of the positive integer $a_n$ is $\widetilde{O}(a_{n-1})$, allowing for poly-logarithmic factors. The conjecture concerns the potential size reduction obtained by using non-monotone formula encodings instead of monotone ones in this restricted arithmetic setting.