Any way to get HD-SDI video from the connect core solutions?

not natively i’m afraid, but something like this will work:
http://www.ti.com/product/lmh0340

or if you need a box solution (pricey):
http://www.amazon.com/HDMI-HD-SDI-Converter-3G-SDI-SD-SDI/dp/B00AHPT98W