Context. GRO J2058+42 is a transient hard X-ray pulsar that occasionally goes into outburst. The optical counterpart is a poorly studied OB-type companion. Aims. We investigate the long-term optical variability of the Be/X-ray binary GRO J2058+42 and the possible connection with periods of enhanced X-ray activity. Methods. We performed an optical spectroscopic and photometric analysis on data collected during about 18 yr. We also present the first optical polarimetric observations of this source. Results. The long-term optical light curves in the BVRI bands and the evolution of the Hα equivalent width display a sinusoidal pattern with maxima and minima that repeat every ∼9.5 yr. The amplitude of this variability increases as the wavelength increases from 0.3 mag in the B band to 0.7 in the I band. The Hα equivalent width varied from about −0.3 to −15 Å. We found a significant decrease in the polarization degree during the low optical state. The intrinsic polarization degree changed by ∼1% from maximum to minimum. The optical maxima occur near periods of enhanced X-ray activity and are followed by a drop in the optical emission. Unlike many other Be/X-ray binaries, GRO J2058+42 does not display V/R variability. Conclusions. The long-term optical variability agrees with the standard model of a Be/X-ray binary, where the circumstellar disk of the Be star grows and dissipates on timescales of 9−10 yr. We find that the dissipation of the disk started after a major X-ray outburst. However, the stability of the Hα line shape as a double-peak profile and the lack of asymmetries suggest the absence of a warped disk and argue against the presence of a highly distorted disk during major X-ray outbursts.

Following the assumption that the disc substructures observed in protoplanetary discs originate from the interaction between the disc and the forming planets embedded therein, we aim to test if these putative planets could represent the progenitors of the currently observed giant exoplanets. We performed N-body simulations initially assuming three, four, five, or seven planets. Our model includes pebble and gas accretion, migration, damping of eccentricities and inclinations, disc-planet interaction, and disc evolution. We located the planets in the positions where the gaps in protoplanetary discs have been observed and we evolved the systems for 100 Myr including a few million years of gas disc evolution, while also testing three values of α viscosity. For planetary systems with initially three and four planets, we find that most of the growing planets lie beyond the radial-velocity (RV) detection limit of 5AU and only a small fraction of them migrate into the inner region. We also find that these systems have final eccentricities that are too low to be in agreement with the observed giant planet population. Systems initially consisting of five or seven planets become unstable after ≈40 Kyr of integration time. This clearly shows that not every gap can host a planet. The general outcome of our simulations – eccentricities that are too low – is independent of the disc’s viscosity and surface density. Further observations could either confirm the existence of an undetected population of wide-orbit giants or exclude the presence of such an undetected population to constrain how many planets hide in gaps even further.