Nonexistence of large-prime lens spaces as dual complexes of log Calabi–Yau fourfolds

Let $p$ be a prime and let $k$ be an integer. Consider a dlt log-Calabi–Yau $4$-fold pair, meaning a dlt pair of dimension $4$ whose log canonical divisor is numerically trivial, and let its dual complex be the cell complex associated to the boundary components and their intersections. Nonexistence conjecture. For sufficiently large prime $p$, there is no dlt log-Calabi–Yau $4$-fold pair whose dual complex is homeomorphic to the lens space $L(p,k)$ for any $k$. This predicts restrictions on which lens spaces can occur as dual complexes of log-Calabi–Yau fourfolds; the examples in the paper produce $L(5,1)$ and $L(5,2)$, while the asserted nonexistence for sufficiently large primes remains open.