  We consider a family of extensions of the Kepler-Coulomb potential on a
$d$-dimensional sphere and analyze it in a deformed supersymmetric framework,
wherein the starting potential is known to exhibit a deformed shape invariance
property. We show that the members of the extended family are also endowed with
such a property, provided some constraint conditions relating the potential
parameters are satisfied, in other words they are conditionally deformed shape
invariant. Since, in the second step of the construction of a partner potential
hierarchy, the constraint conditions change, we impose compatibility conditions
between the two sets to build quasi-exactly solvable potentials with known
ground and first-excited states. Some explicit results are obtained for the
first three members of the family. We then use a generating function method,
wherein the first two superpotentials, the first two partner potentials, and
the first two eigenstates of the starting potential are built from some
generating function $W_+(r)$ [and its accompanying function $W_-(r)$]. From the
results obtained for the latter for the first three family members, we propose
some formulas for $W_{\pm}(r)$ valid for the $m$th family member, depending on
$m+1$ constants $a_0$, $a_1$, \ldots, $a_m$. Such constants satisfy a system of
$m+1$ linear equations. Solving the latter allows us to extend the results up
to the seventh family member and then to formulate a conjecture giving the
general structure of the $a_i$ constants in terms of the parameters of the
problem.
