The homology of a persistence fiber without finite bars

For a finite connected simplicial complex, consider all simplex-wise filters whose ordinary persistent homology has one infinite degree-zero bar born at time 1 and all positive-degree bars born at time 2, with no finite bars. We prove that this fiber has the same homology over $\F$ as the original complex. The fiber has a deformation retract homeomorphic to the order complex of nonempty acyclic subcomplexes of the original complex. Comparing that poset with the face poset gives the homology isomorphism and proves the corresponding Betti-number conjecture of Leygonie and Henselman-Petrusek in the connected setting.