Suppose that $(X,d)$ is a compact metric space. Now suppose that $h:[0,a]\rightarrow[0,b]$ is a continuous function with $h(0)=0$ where if $x\leq y$, then $h(x)\leq h(y)$. Then define $$L(d,h)=\lim_{\epsilon\rightarrow 0}\,\inf\{h(a_{1})+...+h(a_{n})|X=B(x_{1},a_{1})\cup...\cup B(x_{n},a_{n}),$$

$$a_{1}<\epsilon,...,a_{n}<\epsilon,x_{1},...,x_{n}\in X\}.$$

We say that a compact metric space $X$ is a Peano space if it is a compact connected locally connected metrizable space. It is well known that a Hausdorff space is the continuous surjective image of the unit interval if and only if it is a Peano space.

Suppose that $X,Y$ are Peano spaces. Furthermore, assume

whenever $d$ is a metric inducing the topology on $X$ there is a metric $d'$ inducing the topology on $Y$ with $L(d,h)=L(d',h)$ for each monotone mapping $h:[0,a]\rightarrow[0,b]$ with $h(0)=0$ and $x\leq y\Rightarrow h(x)\leq h(y)$, and

whenever $d$ is a metric inducing the topology on $Y$, there is a metric $d'$ inducing the topology on $X$ with $L(d,h)=L(d',h)$ for each monotone mapping $h:[0,a]\rightarrow[0,b]$ with $h(0)=0$ and $x\leq y\Rightarrow h(x)\leq h(y)$.

Then are there nonempty open sets $U\subseteq X,V\subseteq Y$ where $U$ and $V$ are homeomorphic?