  It is known that some well-established parametrizations of the EDF do not
always provide converged results for nuclei and a qualitative link between this
finding and the appearance of finite-size instabilities of SNM near saturation
density when computed within the RPA has been pointed out. We seek for a
quantitative and systematic connection between the impossibility to converge
self-consistent calculations of nuclei and the occurrence of finite-size
instabilities in SNM for the example of scalar-isovector (S=0, T=1)
instabilities of the standard Skyrme EDF. We aim to establish a stability
criterion based on computationally-friendly RPA calculations of SNM that is
independent on the functional form of the EDF and that can be utilized during
the adjustment of its coupling constants. Tuning the coupling constant $C^{\rho
\Delta\rho}_{1}$ of the gradient term that triggers scalar-isovector
instabilities of the standard Skyrme EDF, we find that the occurrence of
instabilities in finite nuclei depends strongly on the numerical scheme used to
solve the self-consistent mean-field equations. The link to instabilities of
SNM is made by extracting the lowest density $\rho_{\text{crit}}$ at which a
pole appears at zero energy in the RPA response function when employing the
critical value of the coupling constant $C^{\rho \Delta\rho}_{1}$ extracted in
nuclei. Our analysis suggests a two-fold stability criterion to avoid
scalar-isovector instabilities: (i) The density $\rho_{\text{min}}$
corresponding to the lowest pole in the RPA response function should be larger
than about 1.2 times the saturation density; (ii) one needs to verify that
$\rho_{p}(q_{\text{pq}})$ exhibits a distinct global minimum and is not a
decreasing function for large transferred momenta.
