Is there a "$\Delta$-isobar puzzle" in the physics of neutron stars?

We discuss the formation of $\Delta$ isobars in neutron star matter. We show that their threshold density strictly correlates with the density derivative of the symmetry energy of nuclear matter, the $L$ parameter. By restricting $L$ to the range of values indicated by recent experimental and theoretical analysis, i.e. $40$ MeV $\lesssim L \lesssim 62$ MeV, we find that $\Delta$ isobars appear at a density of the order of 2$\div$3 times nuclear matter saturation density, i.e. the same range for the appearance of hyperons. The range of values of the couplings of the $\Delta$s with the mesons is restricted by the analysis of the data obtained from photoabsorption, electron and pion scattering on nuclei. If the potential of the $\Delta$ in nuclear matter is close to the one indicated by the experimental data then the equation of state becomes soft enough that a "$\Delta$ puzzle" exists, similar to the "hyperon puzzle" widely discussed in the literature.