Conjecture on connected simple systems of parameterized Conley indices

Let $Z$ be a Hausdorff space, let $\omega\colon M\to Z$ be continuous, and let $P=(P_1,P_2)$ range over regular index pairs for an isolated invariant set $S$. The parameterized Conley index is the ex-space $(I_\omega(P),r_P,s_P)$, where

$$I_\omega(P)=(Z\times 0)\cup(P_1\times 1)\mathbin{/}\sim,$$

with $(x,1)\sim(\omega(x),0)$ for $x\in P_2$, and with the canonical section and projection $s_P$ and $r_P$. Parameterized Conley-index conjecture. The parameterized Conley indices $(I_\omega(P),r_P,s_P)$, as $P$ ranges over all regular index pairs for $S$, form a connected simple system. The corresponding assertion for unparameterized Conley indices is stated as a theorem in the source; the parameterized analogue is proposed as a conjecture.