Forcing extensions where meagre sets are covered

Question

This question fits the Generalised Baire space area. I am interested in the meagre ideal on ${}^\kappa \kappa$, with the bounded topology (or box topology), when, say, $\kappa$ is inaccessible.

To be more precise, basic open sets have the form $[s]=\{t\in {}^\kappa \kappa \mid t\supseteq s\}$, where $s\in {}^{<\kappa}\kappa$. A set $X$ is nowhere dense if every open set has an open subset that does not meet $X$. Finally, $X$ is meagre if it is the $\kappa$-union of meagre sets.

Are there examples or any criterion for a forcing $\mathbb{P}$ to satisfy the following property: "every meagre set in the generic extension is a subset of a meagre set in the sense of the ground model".

Answer 1

This is just a comment, not an answer, but the system does not allow me to comment:

(1) The proof of Claim 5.3(3) in Shelah's 1004 ( https://arxiv.org/abs/1202.5799 ) may as usual give that "old meager sets are cofinal" is the same as "old non-meager sets are non-meager and old reals are dominating"

(2) The A-bounding forcings of Roslanowski/Shelah 860 ( https://arxiv.org/abs/math/0508272 ) and more should fit the bill.

(3) Concerning "preserving non-meagerness" (not related here, but...) I would like to point out the iterable "manageable" condition from section 5 of Matet/Roslanowski/Shelah 799 (https://arxiv.org/abs/math/0210087 )

Andrzej