  The bidirected cut relaxation is the characteristic representative of the
bidirected relaxations ($\mathrm{\mathcal{BCR}}$) which are a well-known class
of equivalent LP-relaxations for the NP-hard Steiner Tree Problem in Graphs
(STP). Although no general approximation algorithm based on
$\mathrm{\mathcal{BCR}}$ with an approximation ratio better than $2$ for STP is
known, it is mostly preferred in integer programming as an implementation of
STP, since there exists a formulation of compact size, which turns out to be
very effective in practice.
  It is known that the integrality gap of $\mathrm{\mathcal{BCR}}$ is at most
$2$, and a long standing open question is whether the integrality gap is less
than $2$ or not. The best lower bound so far is $\frac{36}{31} \approx 1.161$
proven by Byrka et al. [BGRS13]. Based on the work of Chakrabarty et al.
[CDV11] about embedding STP instances into simplices by considering appropriate
dual formulations, we improve on this result by constructing a new class of
instances and showing that their integrality gaps tend at least to $\frac{6}{5}
= 1.2$.
  More precisely, we consider the class of equivalent LP-relaxations
$\mathrm{\mathcal{BCR}}^{+}$, that can be obtained by strengthening
$\mathrm{\mathcal{BCR}}$ by already known straightforward Steiner vertex degree
constraints, and show that the worst case ratio regarding the optimum value
between $\mathrm{\mathcal{BCR}}$ and $\mathrm{\mathcal{BCR}}^{+}$ is at least
$\frac{6}{5}$. Since $\mathrm{\mathcal{BCR}}^{+}$ is a lower bound for the
hypergraphic relaxations ($\mathrm{\mathcal{HYP}}$), another well-known class
of equivalent LP-relaxations on which the current best $(\ln(4) +
\varepsilon)$-approximation algorithm for STP by Byrka et al. [BGRS13] is
based, this worst case ratio also holds for $\mathrm{\mathcal{BCR}}$ and
$\mathrm{\mathcal{HYP}}$.
