The conditional moment closure (CMC) model has successfully been employed for simulation of turbulent nonpremixed combustion in diverse engineering problems. The solution procedure was developed for Lagrangian CMC with a single or multiple fuel or flame groups for a diesel engine in our previous work. Multiple flame groups may consider different histories and residence times through the sequence of evaporation, ignition and mixing controlled combustion. The Lagrangian CMC model has an advantage of less computational burden than Eulerian CMC with fixed spatially resolved grids. However, it cannot allow interaction among different flame groups through premixed combustion, while such interaction occurs through spatial transport terms in Eulerian CMC. In this work an improved version of the Lagrangian CMC model is proposed to consider interaction by premixed flame propagation along a constant mixture fraction contour between neighboring flame groups. The new Lagrangian CMC model is implemented in OpenFOAM and validated for turbulent spray jet flames in diesel-like conditions in the Engine Combustion Network (ECN). Simulations are performed with diverse arrangement of flame groups for a quasi-steady injection strategy in a constant volume chamber. Better agreement is shown by Lagrangian CMC with flame group interaction with measurements in the ECN.