Folklore conjecture on algebraic badly approximable numbers

An irrational real number $x$ is badly approximable if its continued-fraction partial quotients are bounded; write $\mathbf{Bad}$ for the set of badly approximable numbers. An algebraic irrational is an irrational algebraic number.

Folklore conjecture. The only algebraic irrationals that belong to $\mathbf{Bad}$ are the quadratic irrationals.

By the continued-fraction characterization, this predicts that every algebraic irrational of degree at least three has unbounded partial quotients. The problem remains open.