Non-convergence in lexicographic column generation

Hello. i have the following problem with the the following lexicographical model. The master problems look like this:
$C_i^v\) and \(\chi_{itk}^v\) are parameter in the MPs and \(\lambda_{iv}\) and \(g_s\) are variables. 

Master problem first stage:
\begin{align}
&\min \sum_{i\in I}\sum_{v\in V}C_{i}^v\Lambda_{iv}\\
&s.t.\\
&\sum_{i\in I}\sum_{v\in V}\chi_{itk}^v\cdot\Lambda_{iv}\geq Demand_{tk}\quad \forall t\in T,k\in K\\
&\sum_{v\in V}\Lambda_{iv}= 1\quad\forall i\in I\\\
&\Lambda_{iv} \in \left\{ 0;1 \right\}
\end{align}
Master problem second stage:
\begin{align}
&\min \sum_{s\in S }^{}g_s\\\
&s.t.\\
&\sum_{i\in I}\sum_{v\in V}\chi_{itk}^v\cdot\Lambda_{iv}\geq Demand_{tk}\quad \forall t\in T,k\in K\\
&\sum_{v\in V}\Lambda_{iv}= 1\quad\forall i\in I\\\
&g_s \geq \sum_{v\in V}\Lambda_{iv}\cdot C_i^v\quad \forall i \in I_s, s\in S \\
& \sum_{i\in I}\sum_{v\in V}C_{i}^v\Lambda_{iv} = Z^*\\
&\Lambda_{iv} \in \left\{ 0;1 \right\}, g_s \in \mathbb{Z}_{\geq 0} 
\end{align}
And the sub-problems like this:

The duals are \(\pi_{tk}, \eta_i, \mu, \gamma_{si}\), and \(C_i\) (Integer) and \(x_{itk}\) (binary) are variables. 

Subproblem first stage:
\(\min_{v \in V} ~ C_i - \sum_{t \in T} \sum_{k \in K} \pi_{tk} \chi_{itk}^v  -\eta_i\)

Subproblem second stage:
\(\min_{v \in V} ~  - \sum_{t \in T} \sum_{k \in K} \pi_{tk} \chi_{itk}^v-\eta_i  +(-\mu + \gamma_{si})C_i\)

Now I am facing the problem, that in the second stage, the reduced costs don't converge to zero. They will always stay below a negative certain value, f.e. -8.0, but make no progress at all after that. In same model instances, after say 50 iterations, they decrease but wont decrease again for another 1000 iterations.

I observed the following things.

1) After solving the first stage and no more columns are added, i solved this relaxed model again once and found that the duals are non-zero for some constraints. However, when i solve the second-stage (with the previous found columns but none new ones), the duals are zero for every constraints except the new constraint \(\sum_{i\in I}\sum_{v\in V}C_{i}^v\Lambda_{iv} = Z^*\). This value is f.e. 0.5, which effectively reduces the subproblem objective function to \(-0.5C_i\). This incentivizes the SPs to create bigger \(C_i\) than necessary (known from the first stage). Then there is no progress, the duals remain the same and thus, the same column is added over and over again. What could be the problem here? An implementation problem?

If the code is required, i will gladly provide it.