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No, it is not expected to converge. If this is the only source of localization data, the odom->base_link uncertainty will increase forever as time passes. There are different strategies on how to fix this, depending on what info is available.

It is common for robots with a good lidar to correct this odom estimate every few seconds. A lidar-based calculation is used to estimate map->base_link, then map->odom->base_link is corrected, and the odom->base_link covariance is reset. This prevents the odom uncertainty from ever getting very large.