Is the interior of a contractible space contractible if it is path connected?
Take the standard open cone in $\mathbb{R}^3$ (i.e. it does not contain the tip of the cone), about each point on the cone include an (open) line segment normal to that point of length $f(x)$, were $f$ is some function that tends to 0 fast enough so that the resulting space is homeomorphic (in the obvious way) to the open cone crossed with $\mathbb{R}$. Finally, add the cone point.
The space we have described is contractible to the cone point, but its interior is homeomorphic to the open cone crossed with $\mathbb{R}$, hence, noncontractible.