Absence of $^{17}$O Knight-Shift Changes across the First-Order Phase Transition Line in Sr$_2$RuO$_4$

We performed $^{17}$O nuclear magnetic resonance measurements on superconducting (SC) Sr$_2$RuO$_4$ under in-plane magnetic fields. We found that no new signal appears in the SC state and that the $^{17}$O Knight shifts obtained from the double-site measurements remain constant across the first-order phase-transition line, as well as across the second-order phase-transition line as already reported. The present results indicate that the SC spin susceptibility does not decrease in the high-field region, although a magnetization jump in the SC state was reported at low temperatures. Because the spin susceptibility is unchanged in the SC state in Sr$_2$RuO$_4$, we suggest that the first-order phase transition across the upper critical field should be interpreted as a depairing mechanism other than the conventional Pauli-paramagnetic effect.