The best attempt to address this question was by Cury and Uchida (although credit should be given to Verhagen and Wachowiak for the first crack). Cury recorded from mitral cells in freely moving rats while the rats performed odor tasks. During the tasks, the rats switched between normal and fast-breathing, which allowed Cury to compare the neurons' firing during both conditions. They found that the spike timing (or odor code) does not depend on the duration of the breath length (see below; similar to what I blogged about, arguing against a phase-mapping of the sniff cycle). They also noted that during fast breathing, there could be hysteresis, where activity during one breath bleeds into the next.
In the discussion, Cury and Uchida also note the robustness of the odor coding across different breathing regimes. This means the odor code is invariant with respect to inhalation amplitude or duration. In another part of their paper they look at how behaviour relates to odor coding, and found that the responses are relatively insensitive to top-down processing like attention as well.
So, why do many mitral cells only respond to odors after 150ms, which is after the subject has identified the odor? I have two hypotheses.
Hypothesis 1 (why?): The slow breaths do contain more odor information overall, which is contained in the spikes later in the breath. In another figure of the paper, Cury and Uchida show that their odor predictor monotonically increases its accuracy over the entire breathing cycle. However, behavioural data argues against this: when rodents are performing a freely moving odor discrimination task, they fast-sniff.
To test this hypothesis, you could measure a subject's odor discrimination or detection thresholds while it employs a fast- or slow-breathing strategy. If the slow-breathing threshold is lower, it would imply that the late spikes add information. However, if the thresholds are the same (or fast-sniffing lower), then it would argue against the information hypothesis. In the end, this may be difficult to test in rodents, as you need to force them to employ a specific breathing strategy.
Hypothesis 2 (how?): The responses after 150ms are vestigial, due to continued ORN input, or reverberations in the olfactory bulb. ORNs can have complex temporal responses that last for seconds. It is possible that they continue feeding odor-specific information to the olfactoroy bulb after the olfactory bulb no longer needs it. This input would manifest itself as late-arriving responses.
Another possibility is that the olfactory bulb contains recurrent connections, like dendro-dendritic inhibition, that allow the structure to reverberate. For example, stimulating the olfactory nerve can cause mitral cell activity that lasts for seconds. The mitral cells may already have all the information they need after 150ms, but continue firing due to these reverberations.
This hypothesis is more easily testable. You can record from mitral cells in OMP-Halorhodopsin mice. Then during each sniff, you can turn on the light after 150ms, shutting down ORN input to the mitral cells. If the mitral cell activity is elided, then they require ORN input to continuously fire; if, however, the mitral cells continue to fire, then the ORN input is not needed.