Dospinescu–Paș̆kūnas–Schraen's Hecke-eigenspace infinitesimal character conjecture

Let $x:\mathT^S(K^p)[1/p]\to\overline{\mathbf Q}_p$ be a continuous ring homomorphism, let $\mathfrak m_x=\ker(x)$, and suppose that the global Langlands program supplies a corresponding Galois representation $\rho_x:\operatorname{Gal}_{\mathbf Q,S}\to{}^LG_f(\overline{\mathbf Q}_p)$. Let $i\geq0$ be an integer, and let $\zeta_{\rho_x}^C:Z(\mathfrak g)_{\mathbf Q_p}\to\overline{\mathbf Q}_p$ be the character obtained from $\rho_x$ by the Dospinescu–Paș̆kūnas–Schraen construction. Dospinescu–Paș̆kūnas–Schraen's Hecke-eigenspace infinitesimal character conjecture. The algebra $Z(\mathfrak g)_{\mathbf Q_p}$ acts on

$$(\widetilde{\mathbb H}^i[\mathfrak m_x])^{\operatorname{la}}\otimes_{\mathmathT^S(K^p),x}\overline{\mathbf Q}_p$$

via $\zeta_{\rho_x}^C$. This gives a precise Hecke-eigenspace formulation of the expected relation between infinitesimal characters in completed cohomology and the Hodge–Tate–Sen data of the associated Galois representation. The global Langlands construction of $\rho_x$ is itself conditional in the stated generality, and the conjecture is confirmed by the present paper only in its specified setting.