Approximate projective homogeneity conjecture for the pseudo-circle

Let $C$ be a pseudo-circle and $Y$ a planar circle-like continuum. For maps between circle-like continua, write $\deg(f)$ for the integer induced by $f$ on first Čech cohomology. Let $f,g:C\to Y$ be surjections with

$$\deg(f)=\deg(g).$$

A local homeomorphism $\varphi:C\to C$ is a map that is locally a homeomorphism at every point.

Pseudo-circle projective-homogeneity conjecture. For every $\epsilon>0$, there exists a local homeomorphism $\varphi$ of $C$ onto itself such that, for every $x\in C$,

$$|f(x)-g(\varphi(x))|<\epsilon.$$

The conjecture asks for approximate projective homogeneity of the pseudo-circle under the degree condition. The source states it after noting that the pseudo-circle lacks the corresponding property in the setting previously considered; no resolution is supplied here.