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We prove Rubin's conjecture on the structure of local units in theanticyclotomic $\mathbb{Z}_p$-extension of unramified quadratic extension of$\mathbb{Q}_p$ in $p=3$ case by extending Burungale-Kobayashi-Ota's work.

Rubin's conjecture on local units in the anticyclotomic tower at inert primes: $p=3$ case