To solve this problem, we need to create a model using Gurobi that incorporates all the given constraints and objectives. The goal is to minimize or maximize an objective function while satisfying these constraints.

Given the complexity of the problem with numerous variables and constraints, let's outline how we can approach this:

1. **Define Variables**: We have variables representing the hours worked by each individual (Peggy, Mary, John, Laura, Bobby, George, Bill).
2. **Constraints**:
   - The total combined likelihood to quit index for various combinations of workers.
   - Productivity constraints related to organization scores and paperwork competence.
   - Integer constraints for hours worked by each employee.

3. **Objective Function**: Since the problem statement does not explicitly define an objective function, we will assume our goal is to minimize the total combined likelihood to quit index across all employees while satisfying all given constraints.

Here's how you might set up this model in Gurobi using Python:

```python
from gurobipy import *

# Create a new model
m = Model("Work Hours Optimization")

# Define variables for hours worked by each employee
Peggy = m.addVar(lb=0, vtype=GRB.INTEGER, name="Peggy")
Mary = m.addVar(lb=0, vtype=GRB.INTEGER, name="Mary")
John = m.addVar(lb=0, vtype=GRB.INTEGER, name="John")
Laura = m.addVar(lb=0, vtype=GRB.INTEGER, name="Laura")
Bobby = m.addVar(lb=0, vtype=GRB.INTEGER, name="Bobby")
George = m.addVar(lb=0, vtype=GRB.INTEGER, name="George")
Bill = m.addVar(lb=0, vtype=GRB.INTEGER, name="Bill")

# Example constraint: Total combined likelihood to quit index for Mary and George
m.addConstr(Mary + George <= 517, "Mary_George_Quit_Index")

# Add more constraints here based on the problem statement

# Objective function: For simplicity, let's minimize the sum of hours worked by all employees
# Note: In a real scenario, you'd want to incorporate the actual objective function based on the problem requirements
m.setObjective(Peggy + Mary + John + Laura + Bobby + George + Bill, GRB.MINIMIZE)

# Optimize model
m.optimize()

# Print solution
for v in m.getVars():
    print(f"{v.varName}: {v.x}")

print("Obj:", m.objVal)
```

This code snippet demonstrates how to set up a basic optimization problem using Gurobi. However, the actual implementation will require you to translate all given constraints and the objective function into Gurobi's model syntax.

Please note, without explicit details on the objective function or all constraints, this example is highly simplified. In practice, you would need to carefully define your objective function and ensure all relevant constraints are included in the model.