The density Turan problem for 3-uniform linear hypertrees. An efficient testing algorithm

Let $\mathcal{T}=(V,\mathcal{E})$ be a  3-uniform linear hypertree. We consider a blow-up hypergraph $\mathcal{B}[\mathcal{T}]$. We are interested in the following problem. We have to decide whether there exists a blow-up hypergraph $\mathcal{B}[\mathcal{T}]$ of the hypertree $\mathcal{T}$, with hyperedge densities satisfying some conditions, such that the hypertree $\mathcal{T}$ does not appear in a blow-up hypergraph as a transversal. We present an efficient algorithm to decide whether a given set of hyperedge densities ensures the existence of a 3-uniform linear hypertree $\mathcal{T}$ in a blow-up hypergraph $\mathcal{B}[\mathcal{T}]$.