Consider a partially ordering on $\mathbb{R}^n$ that forms a lattice, with meet and join continuous w.r.t. the standard topology (i.e. a topological lattice). Can we choose a path $\gamma(t)$ with arbitrary distinct endpoints also of our choosing $\gamma(0) \leq \gamma(1)$ which preserves order; i.e., $t \leq t' \implies \gamma(t) \leq \gamma(t')$ for any $t$ and $t'$?
I'm pretty sure this is true for Riesz spaces and am wondering if it holds more generally. If the above holds with additional conditions (complete, complemented, distributive, etc) that would also be interesting. It would also be o.k. if $t \leq t' \implies \gamma(t) \leq \gamma(t')$ held in a dense subset of $[0, 1]$.
It feels like this should have been studied somewhere -- I'd appreciate any results or references!
Update: As an attempt, I've been trying to start with an arbitrary path $\gamma(t)$ and using completeness to construct another path $ \bar \gamma(t) = sup_{t' \leq t} \gamma(t') $. But there's no reason to believe that the new $\bar \gamma$ is a continuous path in $\mathbb{R}^n$.
It also seems that a necessary condition to this problem is that we can find an uncountably infinite chain in $\mathbb{R}^n$. This is true for certain partial orders; say, for example, the Riesz space where meet is componentwise min and join is componentwise max. But is it always true for any continuous meet and join operations?
Update 2: I've dug up the following result from "A Compendium of Continuous Lattices" (Gierz et. al.) which is close:
Unfortunately the results in this text rely heavily on compactness. But the result is also stronger than required: it shows that there's an arc-chain (the correct terminology for the above) between any $x < y$, while I'm only looking for the existence of any arc-chain. Perhaps we can find a compact sublattice of $\mathbb{R}^n$.