gn.general topology - Unusual Space-Filling Curve

Around 1998, I encountered a (forgotten) reference to a particularly strange space-filling curve.

Consider a foliation as a collection of continuous nonintersecting curves that start at (0,0) and end at (1,1) and collectively fill the unit square, such as the graphs of functions f_t(x) = x^t where t >=0. Supposedly there exists a continuous curve G that starts at (1,0), ends at (0,1), fills the unit square, and crosses each f_t curve only once.

This initially sounds even more impossible than the Cantor curve. But intuitively a space-filling curve could trace back and forth over the f_t curves and only cross at the corners (0,0) and (1,1). Can someone please explain a construction of such a space-filling curve?