Non-idempotency 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 $f\times f$ for two parallel instances of $f$; an operation is idempotent when $f\times f\equiv_{\mathrm{W}}f$. Idempotency conjecture. Connected choice in dimension two is not idempotent:

$$\text{\rm\sffamily CC}_2\times\text{\rm\sffamily CC}_2\not\equiv_{\mathrm{W}}\text{\rm\sffamily CC}_2.$$

The paper proves that connected choice in dimension one is not idempotent and explains that the conjecture would strengthen the corresponding Brouwer fixed point conjecture. The dimension-two case remains open.