Conjecture on eventual coincidence of the Assouad spectra

Let $F\subseteq\mathbb{R}^d$. Denote by $\overline{\dim}_{\mathrm{A}}^\theta F$ and $\dim_{\mathrm{A}}^\theta F$ the upper and ordinary Assouad spectra, respectively. Eventual coincidence conjecture. For any set $F\subseteq\mathbb{R}^d$, there exists $\theta_0\in(0,1)$ such that

$$\overline{\dim}_{\mathrm{A}}^\theta F=\dim_{\mathrm{A}}^\theta F$$

for all $\theta\in[\theta_0,1)$. The conjecture asserts that the two spectra cannot differ on infinitely many intervals accumulating at $\theta=1$, unlike the known examples of strict decrease on infinitely many intervals accumulating at $\theta=0$.