Wqo for continuous reducibility on real functions

Determine whether continuous reducibility is a well-quasi-order on the class of continuous real functions (functions from the real line to the real line).

Background

The paper contrasts settings where continuous reducibility is known to be bqo (zero-dimensional domains) with settings where related quasi-orders (like embeddability on [0,1]^2) are analytic complete. The status for continuous real functions remains unsettled; resolving the wqo nature would clarify the boundary between tractable and intractable domains for continuous reducibility.