Let's assume all spaces are metrizable. For each connected compact space $X$ let $\mathscr K(X)$ be the set of all partitions of $X$ into non-empty compact sets, excluding the trivial partition $\{X\}$. Let $$\mathfrak \kappa=\min\{|\mathcal K|:(\exists \text{ a connected compactum }X)(\mathcal K\in \mathscr K(X))\}.$$

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

the first inequality is due to Sierpinski (Theorem 6.1.27 in Engelking's Topology), and the second is true because every connected space has a partition into $\mathbb R$-many singletons.

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

EDIT: What about just for the space $X=[0,1]$? Is the minimum cardinality of a compact partition of $X$ necessarily $|\mathbb R|?$