We construct $d\times d$ dimensional bound entangled states, which violate
for any $d>2$ a bipartite Bell inequality introduced in this paper. We
conjecture that the proposed class of Bell inequalities act as dimension
witnesses for bound entangled states: for any $d>2$, there exists a Bell
inequality from this class, which can be violated with bound entangled states
only if their Hilbert space dimension is at least $d\times d$. Numerics
supports this conjecture up to $d=8$.