  We propose a new version of the scalar Weak Gravity Conjecture (WGC) which
would apply to any scalar field coupled to quantum gravity. For a single scalar
it is given by the differential constraint $V''\leq
(2V'''^2/V''-V'''')M_{\text{p}}^2$. It corresponds to the statement that
self-interactions of a scalar must be stronger than gravity for any value of
the scalar field. We find that the solutions which saturate the bound
correspond to towers of extremal states with mass
$m^2(\phi)=m_0^2/((R/m)^2+1/(nR)^2)$, with $R^2=e^\phi$, consistent with the
emergence of an extra dimension at large or small $R$ and the existence of
extended objects (strings). These states act as WGC states for the scalar
$\phi$. It is also consistent with the distance swampland conjecture with a
built-in duality symmetry. From this constraint one can derive several
swampland conjectures. In particular one finds that an axion potential is only
consistent if $f\leq M_{\text{p}}$. The conjecture has far reaching
consequences and applies to several interesting physical systems: i) Among
chaotic inflation potentials only those asymptotically linear may survive. ii)
If applied to the radion of the circle compactification of the Standard Model
to 3D, the constraint implies that the 4D cosmological constant scale must be
larger than the mass of the lightest neutrino, which must be Dirac and in
normal hierarchy. iii) It also constraints simplest moduli fixing string
models. The simplest KKLT model is compatible with the constraints but the
latter may be relevant for some choices of parameters.
