Strong second term identity conjecture

Let $\mathcal{V}^{+}$ and $W$ be the spaces occurring in the paper's regularized Siegel–Weil formula, and let $\mathcal{S}((\mathcal{V}^{+}\otimes W)(\mathbb{A}))$ be the corresponding Schwartz space. The weak second term identity and the weak first term identity on the boundary are identities currently established only for a subspace $\mathcal{S}((\mathcal{V}^{+}\otimes W)(\mathbb{A}))^{\circ}$.

Strong second term identity conjecture. The weak second term identity and the weak first term identity on the boundary can be extended to every

$$\varphi\in\mathcal{S}((\mathcal{V}^{+}\otimes W)(\mathbb{A})).$$

The conjecture asks for the strong form of the identities, removing the restriction to the smaller Schwartz subspace. It is presented as an expected extension of the paper's weak second-term and boundary first-term identities.