Leading term conjecture at $s=r$

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$, let $r>1$ be an integer, and let $p$ be an odd prime. Let $\phi_r$, $A_{\phi_r}^S$, $\psi_r$, $\partial_p$, and $\Omega_{\psi_r,S}$ be the objects defined in the paper. Leading term conjecture at $s=r$. The following assertions hold: (1) $\Sha^1(\mathcal O_{L,S},\mathbb Z_p(r))$ vanishes; (2) $A_{\phi_r}^S\in\zeta(\mathbb Q[G])^\times$; and (3)

$$\partial_p(A_{\phi_r}^S)^\sharp=-\Omega_{\psi_r,S}.$$

These assertions form the paper's central conjecture, combining a Tate–Shafarevich vanishing statement, rationality of normalized leading terms, and an equality of arithmetic invariants; no resolution is stated in the supplied text.