The sequence-growth conjecture for optimal presentations

Let $O$ or $O_{|}$ be the operation set $\{1,S,+,*\}$, and let $(u_k)$ be the sequence of optimal-presentation quantities used in the source. Here $\delta\in\mathbb Z$ is fixed, and $K$ is a sufficiently large constant.

Sequence-growth conjecture. For all $k\in\mathbb N$,

$$(u_k)^2<u_{2k+\delta}$$

for some fixed $\delta\in\mathbb Z$. In particular, for all $k>K$,

$$(u_k)^2<u_{2k+1}.$$

The source presents this as a possible route to improved general upper bounds and a connection with questions in number theory; no resolution is given.