Is it always true that the bid-ask spread sampled before a transaction is smaller than the spread sampled during any other instant between the time duration of these 2 trades?

This may be justified by the fact that traders are more willing to accept crossing the bid ask spread when it is perceived to be less expensive.

for example assume that we have a transaction (occurrence of a trade ) at instant $T_5$. Is the spread observed at this time always smaller than the spread observed during the time frame following the trade $T_4$ and that until the occurrence of the trade $T_5$? Is the sample spread observed $T_5$ the smallest among the sampled spread observed between $T_4$ and $T_5$