Consider a path connected topological space $X$, one can equip its path space $PX=\{ \gamma: [0,1] \longrightarrow X \; continuous\}$ with the compact open topology. We call a ** motion planning algorithm** of $X$, any continuous section $s:X\times X \longrightarrow PX$ of the evaluation $$ev: PX \longrightarrow X\times X, \; \gamma\mapsto (\gamma(0), \gamma(1))$$
M. Farber has shown that such section exists if and only if $X$ is contractible. Hence, we denote $\mathcal{M}(X)$ the space of such sections.

**Question:** Is $\mathcal{M}(X)$ also contractible when $X$ is contractible?

Any remarks or comments are welcome, thank