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This article explains why the call ChannelService.GetInboundLinksForEntityAndLinkType doesn't return a LinkEntityId.
If you ask for links using DataService, you get link objects back with the LinkEntityId property populated, if appropriate. However, if you use ChannelService for the same request, LinkEntityId is always empty.
The output might differ because the data service is looking in the link table while the channel service is looking at the channel structure. However, they should be in sync.
If the calls don't return the same link object, try rebuilding the channel.