The weak Boolean lattice local dimension conjecture

Let $\mathcal{Q}^{n}_{}$ denote the Boolean lattice on $n$ elements, and let $\operatorname{ldim}(P)$ denote the local dimension of a poset $P$. As $n\to\infty$, $\operatorname{ldim}\big(\mathcal{Q}^{n}_{}\big)$ is asymptotic to order $n/\log n$.

Weak Boolean lattice local dimension conjecture.

$$\operatorname{ldim}\big(\mathcal{Q}^{n}_{}\big) = \Theta\big(\frac{n}{\log n}\big).$$

The conjecture would imply an asymptotically stronger bound in terms of the Boolean dimension and could yield a new proof, possibly with an improved constant factor, of the general upper bound for local dimension of finite posets. The source gives no resolution evidence.