Does CyanogenMod Xperia Z3 Dual support LTE on any SIMs? Wiki does not really say word about that: https://wiki.cyanogenmod.org/w/Z3dual_Info

Stock image does support it.

If it doesn't, then what is current progress in context of dual sim?
Always try to look at the specs directly from the manufacturer.  Sony says LTE is supported on some variants of the device.  CyanogenMod doesn't really have anything to do with it, since the radio code is not usually open-source and is not bundled with CM.

GSM Arena is another good place to get device specs.