The tidy-pair vanishing conjecture

Let $(M^{2k+1},F^{2k})$ be a tidy pair, with $i:F\hookrightarrow M$ the inclusion and $i_*:L^2H_k(F,\partial F)\to L^2H_k(M,\partial M)$ the induced map. Tidy-pair vanishing conjecture.

$$i_*:L^2H_k(F,\partial F)\longrightarrow L^2H_k(M,\partial M)\quad\text{is the zero map}.$$

The source describes this as an apparently weaker version of the cocompact action dimension conjecture, in the context of tidy pairs and the preceding $L^2$-homology vanishing implications. Its general validity remains open.