Do these micro sd card reader could work on a nano?

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Most people here will have no experience with the specific hardware, but perhaps you’ll get lucky with that. Barring someone who knows that specific hardware you’d have to find out if this works on something like an ordinary Linux desktop PC. If it works on a desktop PC, then it should work on a Jetson (though you might have to compile a driver as a kernel module). Without knowing the chipset it is hard to guess.

Do note that even if this works on a Jetson (a reasonably high probability) that it wouldn’t necessarily work for use with an external boot device. If you wanted to use this prior to reaching a fully running Linux kernel, then the boot software would also have to have the drivers, and this is a much lower probability for success.