This question arises in connection with this MO question and especially with Sergei Ivanov's wonderful answer, which showed that for any countable set $Q\subset\mathbb{R}^2$ and every closed set $F\subset Q$, there is a closed connected $G\subset\mathbb{R}^2$ with $G\cap Q=F$. (In fact, he makes $G$ path-connected.)

My question is about the extent to which this phenomenon might generalize to higher cardinals, when the Continuum Hypothesis fails. For example, if the continuum $2^\omega$ is very large, then can we hope to handle uncountable sets $Q$ in the way Sergei handled the countable sets, provided that they have size less than the continuum? Or perhaps the best possible is always just the countable sets? Or is this independent of ZFC?

It seems sensible to introduce what seems to be a new cardinal characteristic here. Specifically, let $\kappa$ be the size of the smallest counterexample, that is, the smallest cardinal size of a set $Q\subset\mathbb{R}^2$ having a closed subset $F\subset Q$ for which there is no closed connected $G\subset\mathbb{R}^2$ with $G\cap Q=F$.

Sergei proved that this cardinal $\kappa$ is uncountable, and obviously $\kappa$ is at most the continuum (it is easy to make counterexamples of size continuum), and so $$\omega_1\leq\kappa\leq 2^\omega.$$ So the question is, what can we say about $\kappa$ in ZFC?

If the Continuum Hypothesis holds, of course, then the two endpoints above are identical and so $\kappa=2^\omega$. But is it consistent with ZFC that $$\omega_1\leq \kappa\lt 2^\omega?$$ Perhaps one can achieve particular values of $\kappa$ by forcing? Is the cardinal $\kappa$ related to other well-known cardinal characteristics? Perhaps the value of $\kappa$ is pushed up to the continuum $2^\omega$ by some of the standard forcing axioms?