We present quasi-simultaneous radio (VLA) and X-ray ($Swift$) observations of the neutron star low-mass X-ray binary (NS-LMXB) 1RXS J180408.9$-$342058 (J1804) during its 2015 outburst. We found that the radio jet of J1804 was bright ($232 pm 4 mu$Jy at $10$ GHz) during the initial hard X-ray state, before being quenched by more than an order of magnitude during the soft X-ray state ($19 pm 4 mu$Jy). The source then was undetected in radio (< $13 mu$Jy) as it faded to quiescence. In NS-LMXBs, possible jet quenching has been observed in only three sources and the J1804 jet quenching we show here is the deepest and clearest example to date. Radio observations when the source was fading towards quiescence ($L_X = 10^{34-35}$ erg s$^{-1}$) show that J1804 must follow a steep track in the radio/X-ray luminosity plane with $beta > 0.7$ (where $L_R propto L_X^{beta}$). Few other sources have been studied in this faint regime, but a steep track is inconsistent with the suggested behaviour for the recently identified class of transitional millisecond pulsars. J1804 also shows fainter radio emission at $L_X < 10^{35}$ erg s$^{-1}$ than what is typically observed for accreting millisecond pulsars. This suggests that J1804 is likely not an accreting X-ray or transitional millisecond pulsar.
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