Probing the temperature gradient in the core boundary layer of stars with gravito-inertial modes: the case of KIC$\,$7760680

Aims: We investigate the thermal and chemical structure in the near-core region of stars with a convective core by means of gravito-inertial modes. We do so by determining the probing power of different asteroseismic observables and fitting methodologies. We focus on the case of the B-type star KIC$\,$7760680, rotating at a quarter of its critical rotation velocity. Methods: We compute grids of 1D stellar structure and evolution models for two different prescriptions of the temperature gradient and mixing profile in the near-core region. We determine which of these prescriptions is preferred according to the prograde dipole modes detected in 4-yr $\textit{Kepler}$ photometry of KIC$\,$7760680. We consider different sets of asteroseismic observables and compare the outcomes of the regression problem for a $\chi^2$ and Mahalanobis Distance merit function, where the latter takes into account realistic uncertainties for the theoretical predictions and the former does not. Results: Period spacings of modes with consecutive radial order offer a better diagnostic than mode periods or mode frequencies for asteroseismic modelling of stars revealing only high-order gravito-inertial modes. We find KIC$\,$7760680 to reveal a radiative temperature gradient in models with convective boundary mixing, but less complex models without such mixing are statistically preferred for this rotating star, revealing extremely low vertical envelope mixing. Conclusions: Our results strongly suggest the use of measured individual period spacing values for modes of consecutive radial order as an asteroseismic diagnostic for stellar modelling of B-type pulsators with gravito-inertial modes.