Stick with the 160/60 rear. The front will take a 120/70 no problem, but there have been serious incidents regarding the rear wheel and wrong tyre fitments.
And although the "qualifyer" might be considered a road tyre by the manufacturer, I still stand by my earlier point that you're unlikely to get it up to the temperature it was designed to work at on a public road (or at least only for a few moments). But hey, if it's free then it's not something to sniff at
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