Cube roots and Continued fractions

Question

I once read that it was not known that for any third degree real number if for its continued fraction (c0; c1, c2, c3, ...) all the iterates are bounded or not. Is this still an open question? I looked at the cube root of N > 1, not a perfect cube, and noted if d/n < the cube root of N depends on whether 3 dN^(2/3) /(d^3 N - n^3) has an upper bound or not. I am sure I can get similar results for d/n > the cube root of N. From my point of view this would simplify the question. I would put it on par with trying to get something similar to Pell's equation for square roots. For 2^(1/3) I wrote a simple program that looked at (around) the first 10,000 iterates or so. The largest was 12,737. I found that the average iterate and average square of the iterate was wildly fluctuating. The denominator had 5,329 digits for the last continued fraction it looked at. There may be a relation between the largest iterate and the log of the last denominator, but thre is very slim evidence for that