Cylinder conjecture for connected choice in dimension two

Let \rm\sffamily CC$_2$ denote connected choice for non-empty connected closed subsets of $[0,1]^2$. For a multi-valued operation $f$, write $\operatorname{id}\times f$ for the product with the identity operation, and write $\leq_{\mathrm{sW}}$ for strong Weihrauch reducibility. An operation is a cylinder when $\operatorname{id}\times f\leq_{\mathrm{sW}}f$. Cylinder conjecture. Connected choice in dimension two is not a cylinder:

$$\operatorname{id}\times\text{\rm\sffamily CC}_2\not\leq_{\mathrm{sW}}\text{\rm\sffamily CC}_2.$$

The paper proves that connected choice in dimension one is not a cylinder and notes that dimensions at least three are cylinders via their equivalence with closed choice on the unit interval. Thus dimension two is the only unresolved case.