The $L^2$ signature and intersection-cohomology conjecture for fibred boundary metrics

Let $M$ be a manifold with boundary fibration

$$\partial M\stackrel{\phi}{\longrightarrow}B,$$

and suppose that the metric on $M$ is quasi-isometric near the boundary to

$$ds_c^2=R^{2c}(dR^2+k)+\phi^*ds_B^2,$$

where $R\in[1,\infty)$, $-1\leq c<0$, and $k$ is a symmetric two-tensor on $\partial M$ restricting to a metric on each fiber. Let $[[x]]$ denote the smallest integer strictly greater than $x$. The $L^2$ signature conjecture. For $m$ even,

$$\mathcal{H}^i_{(2)}(M,g_c)\cong\operatorname{Im}\bigl(IH^k_{\underline{\mathfrak{m}}+[[-1/(2c)]]}(X,B)\longrightarrow IH^k_{\overline{\mathfrak{m}}-[[-1/(2c)]]}(X,B)\bigr),$$

and for $m$ odd,

$$\mathcal{H}^i_{(2)}(M,g_c)\cong\operatorname{Im}\bigl(IH^k_{\underline{\mathfrak{m}}+[[(-1/(2c))-(1/2)]]}(X,B)\longrightarrow IH^k_{\overline{\mathfrak{m}}-[[(-1/(2c))-(1/2)]]}(X,B)\bigr).$$

Consequently,

$$L^2\!\operatorname{-}\sigma(M)=\operatorname{sgn}\operatorname{Im}\bigl(H^*(M,\partial M)\longrightarrow H^*(M)\bigr)+\sum_{i=2+2k}^{\infty}\tau_i,$$

where $k=[-1/(2c)]$ when $m$ is even and $k=[(-1/(2c))-(1/2)]$ when $m$ is odd. The conjecture would relate the topological signatures studied in the paper to $L^2$ signatures for a family of complete metrics interpolating between fibred cusp and cylindrical metrics; the source says that it is being proved with Daniel Grieser, but gives no resolution in the supplied text.