  We investigate the possibility of satisfying the $SU(5)$ boundary condition
$\mathbf{Y}^d=\mathbf{Y}^{e\,T}$ at the GUT scale within the renormalizable
$R$-parity conserving Minimal Supersymmetric Standard Model (MSSM). Working in
the super-CKM basis, we consider non-zero flavour off-diagonal entries in the
soft SUSY-breaking mass matrices and the $A$-terms. At the same time, the
diagonal $A$-terms are assumed to be suppressed by the respective Yukawa
couplings. We show that a non-trivial flavour structure of the soft
SUSY-breaking sector can contribute to achieving precise Yukawa coupling
unification for all the three families. However, large non-zero values of the
flavour-violating parameter $(m^2_{dl})_{12}$ lead to a strong tension with the
Lepton Flavour Violating (LFV) observables. Nevertheless, the LFV problem does
not arise when the Yukawa coupling unification requirement is restricted to the
second and third families only. We demonstrate that such a scenario is
consistent with a wide set of experimental constraints, including flavour and
electroweak observables, Higgs physics and the LHC bounds. We also point out
that in order to provide a proper value for the relic density of dark matter,
the lightest neutralino needs to be almost purely bino-like and with mass in
the range of 200$\div$600 GeV. Such a clear experimental prediction makes the
flavour-violating $SU(5)$ Yukawa unification scenario fully testable at the LHC
14 TeV.
