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Context. The nitrogen isotopic ratio is considered an important diagnostictool of the star formation process, and $N_2H^+$ is particularly importantbecause it is directly linked to molecular nitrogen $N_2$. However, theoreticalmodels still lack to provide an exhaustive explanation for the observed$^{14}N/^{15}N$ values.  Aims. Recent theoretical works suggest that the $^{14}N/^{15}N$ behaviour isdominated by two competing reactions that destroy $ N_2H^+$: dissociativerecombination and reaction with CO. When CO is depleted from the gas phase, if$N_2H^+$ recombination rate is lower with respect to the $N^{15}NH^+$ one, therarer isotopologue is destroyed faster. This implies that the $N_2H^+$ isotopicratio in protostars should be lower than the one in prestellar cores, andconsistent with the elemental value of ~440. We aim to test this hypothesis,producing the first sample of $N_2H^+ / N^{15}NH^+$ measurements in low massprotostars.  Methods. We observe the $N_2H^+$ and $N^{15}NH^+$ lowest rotationaltransition towards six young stellar objects in Perseus and Taurus molecularclouds. We model the spectra with a custom python code using a constant$T_{ex}$ approach to fit the observations. We discuss in appendix the validityof this hypothesis. The derived column densities are used to compute thenitrogen isotopic ratio.  Results. Our analysis yields an average of $\rm ^{14}N/^{15}N|_{pro} = 420\pm 15$ in the protostellar sample. This is consistent with the protosolarvalue of 440, and significantly lower than the average value previouslyobtained in a sample of prestellar objects. Conclusions. Our results are inagreement with the hypothesis that, when CO is depleted from the gas-phase,dissociative recombinations with free electrons destroy $N^{15}NH^+$ fasterthan $N_2H^+$, leading to high isotopic ratios in prestellar cores, where CO isfrozen on dust grains.

First sample of $\rm N_2H^+$ nitrogen isotopic ratio measurements in low-mass protostars