Prove that $\{x\in X\mid x\cap\kappa$ is $<\gamma$-closed$\}\in\mathcal I^*$.

Question

Suppose that $\mathcal I$ is a $\kappa$-dense and normal ideal in $X$ (that is $\mathcal P(X)/\mathcal I$ has a dense set of power $\leq\kappa$), where $\kappa\subset X$ and $\kappa>\omega$ is regular. Is this enough to prove $\{x\in X\mid x\cap\kappa$ is $<\gamma$-closed$\}\in\mathcal I^*$, for every $\gamma<\kappa$, where $\mathcal I^*$ is the dual filter of $\mathcal I$ and $A$ is $<\gamma$-closed iff for $x\in[A]^{<\gamma}$, $\bigcup x\in A$. I know this holds when $\mathcal I$ is $\kappa^+$-saturated and normal, $X=\kappa = \lambda^+$ because $\{\xi<\kappa\mid cf(\xi)=cf(\lambda)\}\in\mathcal I^*$ (this was proved by Shelah); and i know that $\kappa$-denseness implies $\kappa^+$-saturation.