Let's assume all spaces are metrizable.

It is a well-known theorem of Sierpinski that a connected compact space cannot be the union of a countable number of disjoint compact subsets. 

For each connected compactum $X$ let $\mathscr K(X)$ be the set of all partitions of $X$ into compact sets. 

Let $$\mathfrak \kappa=\min\{|\mathcal K|:(\exists \text{ connected compactum }X)(\mathcal K\in \mathscr K(X))\}.$$ 

We know $$\aleph_0<\kappa \leq |\mathbb R|.$$

Is it necessarily true that $\kappa=|\mathbb R|$?  Or is this axiom-dependent?