Is running an unpaired t-test on paired data conservative?

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I have data that are repeated measurements. But the biologist did not keep track of the pairing information. It seems to me however that if you run an unpaired t-test on paired data that the worst thing that can happen is a loss of power. In other words it should be a conservative thing to do so I can still reject the null hypothesis of a change in means between the two groups.

Or am I just dreaming and you simply cannot analyze paired data for which the pairing information has been lost?

Any insights at all would be much appreciated, thank you.