Early in their lives, planets endure extreme amounts of ionizing radiation from their host stars. For planets with primordial hydrogen and helium-rich envelopes, this can lead to substantial mass loss. Direct observations of atmospheric escape in young planetary systems can help elucidate this critical stage of planetary evolution. In this work, we search for metastable helium absorption---a tracer of tenuous gas in escaping atmospheres---during transits of three planets orbiting the young solar analogue V1298 Tau. We characterize the stellar helium line using HET/HPF, and find that it evolves substantially on timescales of days to months. The line is stable on hour-long timescales except for one set of spectra taken during the decay phase of a stellar flare, where absoprtion increased with time. Utilizing a beam-shaping diffuser and a narrowband filter centered on the helium feature, we observe four transits with Palomar/WIRC: two partial transits of planet d ($P = 12.4$ days), one partial transit of planet b ($P = 24.1$ days), and one full transit of planet c ($P = 8.2$ days). We do not detect the transit of planet c, and we find no evidence of excess absorption for planet b, with $Delta R_mathrm{b}/R_star<0.019$ in our bandpass. We find a tentative absorption signal for planet d with $Delta R_mathrm{d}/R_star = 0.0205pm0.054$, but the best-fit model requires a substantial (-100$pm$14 min) transit-timing offset on a two-month timescale. Nevertheless, our data suggest that V1298 Tau d may have a high present-day mass-loss rate, making it a priority target for follow-up observations.
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