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Not necessarily, assuming they're correct and it's slipping internally, like they've actually watched it slip, then things can be worse in the smallest/biggest cog if there's a bit of play in the whole assembly and force on the outside pulls things out of line slightly. It wouldn't be the first thing I'd look at, only after ruling out worn teeth. I'd still just buy a new cheap one though.
Slipping and getting worse on the 14 cog but better on the larger cog sounds like a new chain on an old block? If the internals were fubar then it's just as bad in all gears