How does the max-pooling architecture in Collobert and Weston's system perform NER and chunking if it loses positional information in pooling?

I am trying to understand the details of the second architecture described in Natural language processing (almost) from scratch. In the paper it is called the "sentence approach" and extracts features from local windows at each position in the input sentence and then takes a max over word positions in the standard max pooling way. Table 9 in the paper shows part of speech tagging, chunking, and NER results for the max pooling approach. For these tasks the system needs to make a prediction for each word in the sentence. How can the pooling system still make accurate predictions for these tasks while the function it implements is invariant to a reordering of the input words? I suppose it isn't quite invariant since the weights for a single window will produce a different activation if the words in that window get reordered, but once the local window features get computed, their ordering is lost. Is this enough to get the results in table 9? Or do the inputs get augmented in a similar way to what is done for SRL, i.e. each input position gets an extra feature that is the displacement to the position being tagged by the network?

Although I can't find any text mentioning that the input feature augmentation occurs for tasks other than SRL, since the feature vector after pooling is of a fixed size, in order to make predictions for multiple positions, they need to run the network repeatedly. In order to not predict the same thing for all positions, the input needs to change when they run the network again.