(1). Let $D^m$ be the closed $m$-disc in $\mathbb{R}^m$. For each $k$, does the $k$-th configuration space on $D^m$ homotopy equivalent to the $k$-th configuration space on $\mathbb{R}^m$ $$ F(D^m,k)\simeq F(\mathbb{R}^m,k) $$ or not?

(2). Let $M$ be a manifold. For each $k$, does the $k$-th configuration space on $M\times [0,1]$ homotopy equivalent to the $k$-th configuration space on $M\times (0,1)$ $$ F(M\times [0,1],k)\simeq F(M\times(0,1),k) $$ or not?

How to prove these two? Thanks.