Finiteness conjecture for simplicial complexes of type {3,4,5}

A simplicial complex is of type $\{3,4,5\}$ if the link of every $(n-2)$-face is one of the cycles $C_3$, $C_4$, or $C_5$. Finiteness conjecture. In every fixed dimension $n$, there are only finitely many simplicial complexes of type $\{3,4,5\}$. This asserts a finiteness phenomenon for simplicial complexes whose codimension-two links have bounded cycle lengths; the supplied text does not state whether the conjecture has been resolved.