指定タイムスタンプのデータが無い場合、過去ログ探索されてしまう

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仕様のようなのであきらめ。

Wayback – Administrators Manual

Archival URL mode allows replay of all versions captured of a particular URL, by modifying the Timestamp. When an Archival URL Replay request is received for a URL, the Wayback Machine will replay the closest version in time to the Timestamp requested of the particular URL.