Saturation conjecture for ideals extending the nonstationary ideal

Let $\kappa=\nu^+$, and let $\theta$ be a regular infinite cardinal less than $\kappa$ with $\theta\ne\operatorname{cf}(\nu)$. Let $NS_\kappa\vert E^\kappa_\theta$ denote the restriction of the nonstationary ideal on $\kappa$ to $E^\kappa_\theta$, and let $I_\kappa$-$\kappa^+$-saturated mean that the ideal has no antichain of size $\kappa^+$ in its quotient Boolean algebra. Saturation conjecture. No $\kappa$-complete ideal on $\kappa$ extending $NS_\kappa\vert E^\kappa_\theta$ is $I_\kappa$-$\kappa^+$-saturated.

The statement removes the cardinal-arithmetic hypothesis $\kappa=2^\nu$ from the preceding result attributed to Shelah. The supplied text presents it as the paper's motivating conjecture and gives no resolution.