We have examined UV spectra recorded by the Space Telescope Imaging Spectrograph (STIS) on the Hubble Space Telescope for three stars, HD32309, 41 Ari, and $eta$~Tel, that are located well inside the boundary of the Local Hot Bubble in our search for absorption features of Si IV, C IV, and N V that could reveal the presence of an interface between the local warm ($Tsim 7000$ K) neutral medium and a more distant hot ($Tsim 10^6$ K) interstellar medium. In all cases, we failed to detect such ions. Our most meaningful upper limit is that for log N(C IV)< 11.86 toward HD32309, which is below the expectation for a sight line that penetrates either a conductive/evaporative interface or a turbulent mixing layer. We offer conjectures on the reasons for these negative results in terms of either a suppression of a conductive layer caused by the shielding of the local cloud by other clouds, which may make it more difficult for us to sense discrete absorption features from gases at intermediate temperatures, or by the presence of a tangential magnetic field at most locations on the surface of the local cloud.
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