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The drivers of compositionality in artificial languages that emerge when two (or more) agents play a non-visual referential game has been previously investigated using approaches based on the REINFORCE algorithm and the (Neural) Iterated Learning Model. Following the more recent introduction of the \textit{Straight-Through Gumbel-Softmax} (ST-GS) approach, this paper investigates to what extent the drivers of compositionality identified so far in the field apply in the ST-GS context and to what extent do they translate into (emergent) systematic generalisation abilities, when playing a visual referential game. Compositionality and the generalisation abilities of the emergent languages are assessed using topographic similarity and zero-shot compositional tests. Firstly, we provide evidence that the test-train split strategy significantly impacts the zero-shot compositional tests when dealing with visual stimuli, whilst it does not when dealing with symbolic ones. Secondly, empirical evidence shows that using the ST-GS approach with small batch sizes and an overcomplete communication channel improves compositionality in the emerging languages. Nevertheless, while shown robust with symbolic stimuli, the effect of the batch size is not so clear-cut when dealing with visual stimuli. Our results also show that not all overcomplete communication channels are created equal. Indeed, while increasing the maximum sentence length is found to be beneficial to further both compositionality and generalisation abilities, increasing the vocabulary size is found detrimental. Finally, a lack of correlation between the language compositionality at training-time and the agents' generalisation abilities is observed in the context of discriminative referential games with visual stimuli. This is similar to previous observations in the field using the generative variant with symbolic stimuli.