Let $I$ be a $\mathrm{Top}_*$-enriched poset and $X: I \to \mathrm{Top}_*$, and consider the homotopy limit $$ \underset{i \in I}{\mathrm{holim}}X(i), $$ where the maps $X(i) \to X(j)$ are nullhomotopic for $i \leq j$ and $X(i) \to X(j)$ a weak homotopy equivalence whenever $i \cong j$.

Can we conclude that $\underset{i \in I}{\mathrm{holim}}X(i)$ is trivial?

## An example

The example I have in mind is whenever $I$ is the poset of non-zero subspaces of $\mathbb{R}^n$ (topologized as a disjoint union of Grassmannians), and $X$ is the sphere functor $\mathbb{S}: V \mapsto S^V$. Then when $\dim V < \dim W$ the map $S^V \to S^W$ is null homotopic and when $\dim V = \dim W$ the map $S^V \to S^W$ is a weak homotopy equivalence.

I'd like to conclude that the homotopy limit, $$ \underset{0 \neq U \subseteq \mathbb{R}^n}{\mathrm{holim}}S^U $$ is weakly contractible.