The morphic-sequence complexity spectrum conjecture

Let $f53f$ be a morphic sequence, and let $p_{\mathbf{w}}(n)$ denote its factor-complexity function. Morphic-sequence complexity spectrum conjecture. The function $p_{\mathbf{w}}$ only has one of the following asymptotic behaviors: $\Theta(1)$, $\Theta(n)$, $\Theta(n\log\log n)$, $\Theta(n\log n)$, or $\Theta(n^{1+\frac{1}{k}})$ for some $k\in\mathbb{N}$. The surrounding text explains that this would extend the known restrictions on morphic-sequence complexity beyond the proved alternatives, while the source gives no resolution of the conjecture.