Let $X$ and $Y$ be topological spaces. A map $f:X\to Y$ belongs to *the first Baire class* (to *the first homotopic Baire class*), if there exists a continuous map $H:X\times \omega\to Y$ (a continuous map $H:X\times [0,+\infty)\to Y$) such that
\begin{gather}
\lim_{n\to\infty} H(x,n)=f(x)
\end{gather}
for every $x\in X$. The collection of all (homotopic) Baire-one maps between $X$ and $Y$ we denote by ${\rm B}_1(X,Y)$ (${\rm hB}_1(X,Y)$).

Evidently, ${\rm hB}_1(X,Y)\subseteq {\rm B}_1(X,Y)$ for any spaces $X$, $Y$, and ${\rm hB}_1(X,Y)={\rm B}_1(X,Y)$ if either $X$ or $Y$ is contractible. Moreover, ${\rm hB}_1(X,Y)={\rm B}_1(X,Y)$ if $X$ is a topological space and $Y$ is a metrizable ANR.

Do there exist path-connected subspaces $X,Y\subseteq\mathbb R^2$ such that ${\rm hB}_1(X,Y)\ne {\rm B}_1(X,Y)$?