Do lazy constraints cut fractional solutions?

Here's a log of my model. I'm just working with the root node (node limit = 1) and have the heuristics parameter set to 0 as well as every gurobi cut (hence the lack of integer solutions) For the sake of simplicity, let \(f_{ij}\) be the variable that considers the amount of flow traversing from \(i\) to \(j\) and let \(d_{ij}\) be a constant amount that considers the demand originated in \(i\) and with destination in \(j\), then, the following constraint must hold: \[\sum_{i \in S, j \in S^c}d_{ij} - \sum_{i \in S, j \in S^c}f_{ij} \leq 0\].

Root relaxation: objective 5.688659e+04, 1279 iterations, 0.04 seconds (0.07 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0 56886.5870    0  103          - 56886.5870      -     -    0s
Violated Demand Constraint found!
    S:{0, 1, 2, 3, 4, 5, 8}    Sc:{9, 6, 7}
    Total demand:21.00    Total flow:8.40

     0     0 56985.9417    0  107          - 56985.9417      -     -    0s
Violated Demand Constraint found!
    S:{0, 1, 2, 3, 4, 5, 8}    Sc:{9, 6, 7}
    Total demand:21.00    Total flow:8.73

     0     0 65344.4268    0   67          - 65344.4268      -     -    0s
Violated Demand Constraint found!
    S:{1, 2, 3, 4, 5, 7, 8}    Sc:{0, 9, 6}
    Total demand:21.00    Total flow:14.39

     0     0 65344.4268    0   67          - 65344.4268      -     -    0s
Violated Demand Constraint found!
    S:{1, 2, 3, 4, 5, 7, 8}    Sc:{0, 9, 6}
    Total demand:21.00    Total flow:14.39

Violated Demand Constraint found!
    S:{1, 2, 3, 4, 5, 7, 8}    Sc:{0, 9, 6}
    Total demand:21.00    Total flow:14.39

     0     2 65344.4268    0   67          - 65344.4268      -     -    0s

Cutting planes:
  Lazy constraints: 4

Explored 1 nodes (2459 simplex iterations) in 0.71 seconds (0.37 work units)
Thread count was 16 (of 16 available processors)

Solution count 0

I'm working at {where=MIPNODE} Let's take the first two lazy constraints "added". \(S\) and \(S^c\) are the same which doesn't make sense since the first lazy constraint states that for that specific \(S,S^c\) the amount of total flow should be \(21\) and so, the second lazy constraint shouldn't even appear there.

Are lazy constraints activated in the relaxation or only taken into consideration whenever an integer solution is found? Should I add these as Cuts as well as LazyConstraints when i'm at MIPNODE? They are 100% needed for the model to be feasible.