The textbook counterexamples of connected Lie subgroups that are not closed seem to all be constructed by having one-parameter subgroups not being closed, e.g. the irrational winding of the Torus. I've seen this question, which has led me to here with some more examples. This reference seems to weaken the requirement of simple connectivity on $G$ for when the connected Lie subgroup associated with an ideal is closed.
However, it is still unclear to me if there are any counterexamples to the following statement, which is inspired by the irrational flow on a Torus example. Consider Lie subalgebra $\mathfrak{h}$ of $\mathfrak{g} \subset \mathfrak{gl}(\mathbb{C}, n)$, where $G$ is a Lie group with algebra $\mathfrak{g}$. Suppose that $G$ is at least connected. Suppose $\mathfrak{h}$ has as a basis $\{ H_{1}, \dots , H_{k} \}$ where each of the one-parameter subgroups associated with a basis element $H_i$ is closed. Does this imply that the connected Lie subgroup associated with $\mathfrak{h}$ is closed? If this is not true in general, are there any additional conditions that can be put on $G$ or $\mathfrak{g}$ to make it true (without any additional conditions on $\mathfrak{h}$)?