  We conjecture that in $\mathcal{N}=1$ supersymmetric 4d string vacua with
non-vanishing gravitino mass, the limit $m_{3/2}\rightarrow 0$ is at infinite
distance. In particular one can write $M_{\text{tower}} \simeq m_{3/2}^\delta$
so that as the gravitino mass goes to zero, a tower of KK states as well as
emergent strings become tensionless. This conjecture may be motivated from the
Weak Gravity Conjecture as applied to strings and membranes and implies in turn
the AdS distance conjecture. We test this proposal in classical 4d type IIA
orientifold vacua in which one obtains a range of values $\tfrac13 \le \delta
\le 1$. The parameter $\delta$ is related to the scale decoupling exponent in
AdS vacua and to the $\alpha$ exponent in the SDC for the type IIA complex
structure. We present a general analysis of the gravitino mass in the limits of
moduli space in terms of limiting Mixed Hodge Structures and study in some
detail the case of two moduli F-theory settings, which yield a lower bound
$\delta>1/4$. The conjecture has important phenomenological implications. In
particular we argue that low-energy supersymmetry of order 1 TeV is only
obtained if there is a tower of KK states at an intermediate scale, of order
$10^8\text{-}10^{13}$ GeV. One also has an upper bound for the Hubble constant
upon inflation $H\lesssim m_{3/2}^\delta M^{(1-\delta)}_{\text{P}}$.
