Consistency of separating the generalized null ideal from Cohen forcing

Let $\bo\mathbf V$ be a model and let $\bo\kappa$ be a sufficiently strong cardinal. Write $\bo\mathbb Q_\kappa$ for the forcing notion associated with the generalized null ideal and $\bo\operatorname{Cohen}_\kappa$ for the forcing adding a $\kappa$-Cohen real. The cardinal $\bo\operatorname{add}(\mathbb Q_\kappa)$ is the additivity of the corresponding ideal, and $\bo\mathfrak b_\kappa$ is the generalized bounding number.

Consistency conjecture. There exists a model $\bo\mathbf V$ such that

$$\bo\mathbf V\models\operatorname{add}(\mathbb Q_\kappa)>\operatorname{add}(\operatorname{Cohen}_\kappa).$$

Since necessarily

$$\bo\mathbf V\models\mathfrak b_\kappa=\operatorname{add}(\operatorname{Cohen}_\kappa),$$

this is equivalently the consistency assertion

$$\operatorname{CON}\bigl(\operatorname{add}(\mathbb Q_\kappa)>\mathfrak b_\kappa\bigr).$$

The claim asks whether the Bartoszyński–Raisonnier–Stern inequality can consistently fail for a sufficiently strong cardinal. The supplied text gives no resolution.