Golden Gait Inc. & Woodke's Walnut Woods Curly Horses & Ponies

I have looked at a lot of sites, to find out more about this lovely breed of horse, there is a lot of information out there, but there is one question that interests me, I have found no answer, so wondered if you would provide that answer, could a straight haired curly produce a curly foal, when put to a non curly, or do they not have the curly gene, even if they are hypoallergenic.