Bounded gap between lower and upper threshold functions

Let $G_{n,p,d}$ denote the class of $d$-dependent random graph distributions on graphs with $n$ vertices and marginal edge probability $p$. Let $\mathcal{P}$ be a monotone graph property, and let $L(n,d)$ and $U(n,d)$ be the lower and upper threshold functions from the existence-of-threshold-functions conjecture. Bounded threshold-gap conjecture. For every monotone graph property $\mathcal{P}$, there exists a function $f(d)$ depending only on $d$ such that

$$\frac{U(n,d)}{L(n,d)}=O(f(d)).$$

If true, the range between the two threshold functions would be controlled solely by the dependence parameter $d$, rather than by $n$. The source states this as an expected open phenomenon; the supplied parser status is unknown.