Relative dimension controlled by block dimensions

Determine whether there exists a function f such that, for every poset P whose blocks all have relative dimension at most d, the relative dimension of P is at most f(d).

Background

For posets with connected cover graphs, a block is the subposet induced by a maximal 2-connected subset of the cover graph. The paper recalls that if every block has dimension at most d, then the whole poset has dimension at most d+2, whereas no analogous boundedness statement holds for local dimension. It remains unresolved whether relative dimension admits some function-of-d control from the relative dimensions of the blocks.