The standardness conjecture for the census 4-ball $B^4_C$

Let $C$ be the specified 2-pentachoron cut-open Cappell--Shaneson complex, and let $B^4_C$ be the topological 4-ball containing $C$ described in the paper. Standardness conjecture for $B^4_C$. $B^4_C$ is PL-homeomorphic to the standard 4-ball. This claim concerns a potential combinatorial obstruction to simplifying triangulations; the supplied text presents it among the paper's results and conjectures, but gives no resolution.