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Voyager 1/UVS measurement of the Lyman α intensities at the distant heliosphere is a unique source of information on the interstellar hydrogen distribution at the heliospheric boundary. In 2003-2014 (at 90-130 AU from the Sun) Voyager 1 measured the Lyman α emission at an almost fixed direction close to the upwind (i.e. toward the interstellar flow). The data show an unexpected behavior in 2003-2009: the ratio of observed intensity to the solar Lyman α flux at Earth orbit is almost constant. We performed a numerical modeling of these data in the frame of a state-of-the-art self-consistent kinetic-MHD model of the heliospheric interface. The model results, for various interstellar parameters, predict a monotonic decrease of intensity not seen in the data. We propose two possible scenarios that explain the data qualitatively. The first is the formation of a dense layer of hydrogen atoms near the heliopause. Such a layer would provide an additional backscattered Doppler-shifted Lyman α emission, which is not absorbed inside the heliosphere and may be observed by Voyager. About 35 R of intensity from the layer is needed. The second scenario is an external nonheliospheric Lyman α component, which could have a galactic or extragalactic origin. Our parametric study shows that about 25 R of additional emission would be enough to provide a good qualitative agreement between the Voyager 1 data and the model results.