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The two-tone dial is a cool design, even if is refinished. I'd also be interested to see more information on the case. And, I'd argue you could still call it a sector dial if you like. It's not what first comes to mind for a sector dial, but it has concentric circles and sectors, albeit small, between hours markers and minute track.
finding a steel case of a 1930s design that would fit a caliber 12.92 seems a difficult task.
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According to Longines: "It was invoiced on 16 May 1935 to the company Hattori, which was at that time our agent for Japan".
I purchased the watch from a Japanese seller.