On the $l_2$-stability of the spaces of homeomorphisms of metric spaces

The space $H(M)$ of homeomorphisms of a metric space $M$ is homeomorphic to the product $H(M)\times l_2$ if there exists an embedding $\varphi\colon K\times[0,1]\to M$ of the product of a metrizable compact space $K$ and the segment $[0,1]$ into the space $M$ such that $\operatorname{Int}\varphi(K\times[0,1])\ne\varnothing$.