Compatibility of the plectic action with Poincaré duality

Establish that the subspace \(Y\) arising in the Poincaré-duality argument is stable under the constructed algebra \(A^{(r)}\), thereby proving compatibility between the plectic symmetry action and the Poincaré duality spectral sequence.

Background

The proof of the below-middle-degree annihilation theorem switches from the constructed plectic algebra $A^{(r)}$-action to the smaller $R_{\bar D}[G_Q]$-action because the authors cannot establish the required compatibility with Poincaré duality.

In the argument, $Y$ is defined as an intersection involving the image of a Poincaré-duality morphism. Stability of $Y$ under $A^{(r)}$ would allow the proof to retain the full plectic algebra throughout, but this stability is not proved.