Approximate projective homogeneity conjecture for pseudo-solenoids

Let $C$ be a pseudo-solenoid and let $Y$ be a circle-like continuum with the same first Čech cohomology groups as $C$. Let $f,g:C\to Y$ be shape equivalences with equal degree, where the degree is the induced integer on first Čech cohomology. A homeomorphism $\varphi:C\to C$ is a bijective homeomorphism of $C$ onto itself.

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

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

This is the pseudo-solenoid analogue of the preceding projective-homogeneity claim and is presented as a conjecture for future Fraïssé-limit constructions. The source gives no resolution.