The asymptotic local dimension conjecture for powers of chains

Let $\mathbf{t}^n$ denote the $n$-fold product of a chain with $t$ elements, let $\operatorname{ldim}(P)$ denote the local dimension of a poset $P$, and let $\log_t n$ be the logarithm of $n$ to base $t$. There should be a universal constant $c$ independent of $t$ such that, for every $t\in\mathbb{N}$, the following asymptotic holds as $n\to\infty$.

Asymptotic local dimension conjecture for powers of chains. There exists a universal constant $c$ such that, for any $t\in\operatorname{\mathbb{N}}$, as $n\to\infty$,

$$\operatorname{ldim}\big(\mathbf{t}^n\big) \sim c\frac{n}{\log_t n}.$$

This is presented as a stronger conjecture than the Boolean-lattice claim, extending the proposed asymptotic to products of chains. The source gives no resolution evidence.