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We study the emission from a molecular photonic cavity formed by two proximal photonic crystal defect cavities containing a small number (<3) of In(Ga)As quantum dots. Under strong excitation we observe photoluminescence from the bonding and antibonding modes in excellent agreement with expectations from numerical simulations. Power dependent measurements reveal an unexpected peak, emerging at an energy between the bonding and antibonding modes of the molecule. Temperature dependent measurements show that this unexpected feature is photonic in origin. Time-resolved measurements show the emergent peak exhibits a lifetime $\tau_M=0.75 \, \pm 0.1 \, ns $, similar to both bonding and antibonding coupled modes. Comparison of experimental results with theoretical expectations reveal that this new feature arises from a coexistence of weak- and strong-coupling, due to the molecule emitting in an environment whose configuration permits or, on the contrary, impedes its strong-coupling. This scenario is reproduced theoretically for our particular geometry with a master equation reduced to the key ingredients of its dynamics. Excellent qualitative agreement is obtained between experiment and theory, showing how solid-state cavity QED can reveal new regimes of light-matter interaction.