Local weak contractibility conjecture for pseudo-isotopy spaces

Let $f\taut$ be a map, where $\taut$ denotes the boundary of the space of embeddings, and let $\tP(f)$ denote the corresponding pseudo-isotopy space. Local weak contractibility conjecture. Every path component of $\tP(f)$ is weakly contractible: for every $k>0$, any map $S^k\to\tP(f)$ extends to a map $D^{k+1}\to\tP(f)$. The conjecture is motivated by local contractibility of the group of orientation-preserving homeomorphisms of $S^2$; the source indicates that the proposed argument has not been pursued.