Open problems in continued fractions theory

Zaremba's Conjecture:

every integer appears as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant.

From M. Waldschmidt, "Open Diophantine Problems" (Moscow Mathematical Journal vol. 4, no. 1, 2004, pp. 245-305):

• Does there exist a real algebraic number of degree $\geq 3$ with bounded partial quotients?
• Does there exist a real algebraic number of degree $\geq 3$ with unbounded partial quotients?

Guy, Unsolved Problems In Number Theory, F21, attributes to Bohuslav Divis the conjecture that in each real quadratic field there is an irrational with all partial quotients 1 or 2; more generally, same question but with 1 and 2 replaced by any pair of distinct positive integers.