Suppose that $M$ is a compact PL-manifold (possibly with boundary) and let $C^{PL}(M)$ denote the (simplicial) group of PL isomorphisms of $M \times I$ relative to $M \times \{0\} \cup \partial M \times I$, i.e. $PL(M\times I,M \times \{0\} \cup \partial M \times I)$. There is a stabilization map $\sigma: C^{PL}(M) \to C^{PL}(M \times I)$. A hypothetical PL-pseudoisotopy stability would be:

There is a function $f(k)$ so that the map $\sigma: C^{PL}(M) \to C^{PL}(M \times I)$ is $k$-connected for $n = \dim(M) \geq f(k)$.

In other words, the homotopy groups of the PL-pseudoisotopy space stabilize as one multiplies with intervals.

The history of this theorem is as follows as far as I understand. Hatcher in 'Higher Simple Homotopy Theory' gives an outline of a possible proof of this with $f(k)$ approximately $3k$. In 1988, following Hatcher's outline for the PL-case, Igusa gave a very detailed proof for the smooth case. If $M$ is a compact smooth manifold, $C^{DIFF}(M)$ is the topological (or simplicial) group $DIFF(M\times I,M \times \{0\} \cup \partial M \times I)$ and $\sigma$ again the stabilization map, then the statement of the smooth pseudoisotopy stability theorem is as follows:

Let $n = \dim(M)$. The stabilization map $\sigma: C^{DIFF}(M) \to C^{DIFF}(M \times I)$ is $k$-connected for $n\geq\max(2k+7, 3k+4)$.

This implies, by an argument of Burghulea and Goodwillie, a PL-pseudoisotopy stability theorem for smoothable PL-manifolds M with the same range as Igusa's theorem. Also a general PL-pseudoisotopy stability theorem by smoothing theory implies the smooth pseudoisotopy stability theorem. However, as far as I can find the general PL case is still open (see e.g. Waldhausen-Jahren-Rognes, page 22). My questions are thus as follows:

• Is the general PL-pseudoisotopy stability theorem still open? If not, is there a reference for a detailed proof?
• If the answer to the previous question is yes, do people expect the PL-pseudoisotopy stability to be true? Is the range expected to the be same as Igusa's?
• If the answer to the previous question is yes, how is a hypothetical proof expected to go? Is it a matter of making Hatcher's outline precise to get something similar to Igusa's proof or are new ideas needed?