Strictness conjecture for connected choice in dimension two versus interval choice

Let \rm\sffamily CC$_2$ be connected choice for non-empty connected closed subsets of $[0,1]^2$, and let \rm\sffamily C$_{[0,1]}$ be closed choice on the unit interval. Weihrauch reducibility is written $\leq_{\mathrm{W}}$, with strict reducibility written $<_{\mathrm{W}}$. Dimension-two connected-choice conjecture. Connected choice in dimension two is strictly weaker than closed choice on the unit interval:

$$\text{\rm\sffamily CC}_2<_{\mathrm{W}}\text{\rm\sffamily C}_{[0,1]}.$$

This is the paper's stated major open problem: whether closed choice on the interval reduces to connected choice in dimension two. The conjecture asserts that it does not, while the reverse reduction is immediate from the definitions.