  We investigate Gamma conjecture I and its underlying Conjecture $\mathcal{O}$
for the $\mathbb{P}^1$-bundles
$X_n=\mathbb{P}_{\mathbb{P}^{n}}(\mathcal{O}\oplus\mathcal{O}(n))$ with $n\ge
3$. We show that Conjecture $\mathcal{O}$ does not hold if $n$ is odd, and that
Gamma conjecture I does not hold if $n$ is even. Led by this example, we
propose modifications for Gamma conjecture I, discuss Gamma conjecture I over
the Kahler moduli space, and identify the corresponding principal asymptotic
class.
