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Frequent, low-latency measurements of the Earth's rotation phase, UT1$-$UTC, critically support the current estimate and short-term prediction of this highly variable Earth Orientation Parameter (EOP). Very Long Baseline Interferometry (VLBI) Intensive sessions provide the required data. However, the Intensive UT1$-$UTC measurement accuracy depends on the accuracy of numerous models, including the VLBI station position. Intensives observed with the Maunakea (Mk) and Pie Town (Pt) stations of the Very Long Baseline Array (VLBA) illustrate how a geologic event (i.e., the $M_w$ 6.9 Hawai`i Earthquake of May 4th, 2018) can cause a station displacement and an associated offset in the values of UT1$-$UTC measured by that baseline, rendering the data from the series useless until it is corrected. Using the non-parametric Nadaraya-Watson estimator to smooth the measured UT1$-$UTC values before and after the earthquake, we calculate the offset in the measurement to be 75.7 $\pm$ 4.6 $μ$s. Analysis of the sensitivity of the Mk-Pt baseline's UT1$-$UTC measurement to station position changes shows that the measured offset is consistent with the 67.2 $\pm$ 5.9 $μ$s expected offset based on the 12.4 $\pm$ 0.6 mm total coseismic displacement of the Maunakea VLBA station determined from the displacement of the co-located global navigation satellite system (GNSS) station. GNSS station position information is known with a latency on the order of tens of hours, and thus can be used to correct the a priori position model of a co-located VLBI station such that it can continue to provide accurate measurements of the critical EOP UT1$-$UTC as part of Intensive sessions. The VLBI station position model would likely not be updated for several months. This contrast highlights the benefit of co-located GNSS and VLBI stations in support of the monitoring of UT1$-$UTC with single baseline Intensives. Abridged.