  It is known that the Real Extended Bialas-Bzdak (ReBB) model describes the
proton-proton ($pp$) and proton-antiproton ($p\bar p$) differential
cross-section data in a statistically non-excludible way,\linebreak i.e., with
a confidence level greater than or equal to 0.1\% in the center of mass energy
range \linebreak 546 GeV $\leq\sqrt{s}\leq$ 8 TeV and in the squared
four-momentum transfer range 0.37 GeV$^2$ $\leq -t\leq$ 1.2 GeV$^2$.
Considering, instead of Gaussian, a more general L\'evy $\alpha$-stable shape
for the parton distributions of the constituent quark and diquark inside the
proton and for the relative separation between them, a generalized description
of data is obtained, where the ReBB model corresponds to the $\alpha =$ 2
special case. Extending the model to $\alpha <$ 2, we conjecture that the
validity of the model can be extended to a wider kinematic range, in
particular, to lower values of the four-momentum transfer $-t$. We present the
formal L\'evy $\alpha$-stable generalization of the Bialas-Bzdak model and show
that a simplified version of this model can be successfully fitted, with
$\alpha<$ 2, to the non-exponential, low $-t$ differential cross-section data
of elastic proton-proton scattering at $\sqrt{s} =$ 8 TeV.
