Remedy for NH34 GMT hand not disengaging?

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I have an inexpensive watch fitted with the NH34 and a screw-down crown. After setting the GMT hand, I depress the crown to screw it in. The GMT hand is still engaged and turns when I screw the crown down.

I wonder if there's something up with the stem… too long? Too short?!

Or maybe the tube is sitting a bit proud of the case so that the crown can't be depressed enough (prior to screwing down) to disengage the GMT hand.