LS~5039 is a powerful gamma-ray binary that probably hosts a non-accreting pulsar. Despite the wealth of data available, the power source of the non-thermal emitter is still unknown. We use a dynamical-radiative numerical model and multiwavelength data to constrain the properties of a pulsar wind that may power the non-thermal emitter in LS~5039. We ran simulations of an ultrarelativistic (low-$B$) cold $e^pm$-wind that Compton scatters stellar photons and that dynamically interacts with the stellar wind. The effects of energy losses on the unshocked $e^pm$-wind dynamics, and the geometry of the two-wind contact discontinuity, are computed for different wind models. The predicted unshocked $e^pm$-wind radiation at periastron, when expected to be highest, is compared to LS~5039 data. The minimum possible radiation from an isotropic cold $e^pm$-wind overpredicts the X-ray to gamma-ray fluxes at periastron by a factor of $sim 3$. In the anisotropic wind case X-ray and $gtrsim 100$ MeV data are not violated by wind radiation if the wind axis is at $lesssim 20-40^circ$ from the line of sight (probability of $lesssim 6-24$%), depending on the anisotropic wind model, or if the wind Lorentz factor $in 10^2-10^3$, in which case the wind power can be higher, but it requires $e^pm$-multiplicities of $sim 10^6$ and $10^9$ for a $10^{-2}$~s and 10~s pulsar period, respectively. The studied model predicts that a low-$B$ cold pulsar $e^pm$-wind in LS~5039 should be strongly anisotropic, with either a wind Lorentz factor $in 10^2-10^3$ and very high multiplicities or with a fine-tuned wind orientation. A low-$B$, cold baryon-dominated wind would be possible, but then the multiplicities should be rather low, while the baryon-to-$e^pm$ energy transfer should be very efficient at wind termination. A strongly magnetized cold wind seems to be the most favorable (least constrained) option.
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