Assume that $X$ is a topological space. We say that $X$ satisfies the strong fixed point property if the graph of every surjective continuous self-map intersect the graph of every continuous self-map on $X$. For example the interval $I=[0,1]$ satisfy this property.
An equivalent definition: If two continuous self-maps $f,g$ on $X$ have non intersecting graphs, then neither $f$ nor $g$ is surjective.
Are there some examples of manifolds (with or without boundary) of higher dimension with this property? In particular, do the closed $2$_disc, or the even dimensional real or complex projective spaces satisfy this property?