  We introduce a three-parameter family of Bell functionals that extends those
studied in reference [Phys. Rev. Research 2, 033420 (2020)] by including a
marginal contribution. An analysis of their quantum value naturally splits the
family into two branches, and for the first of them we show that this value is
given by a simple function of the parameters defining the functionals. In this
case we completely characterise the realisations attaining the optimal value
and show that these functionals can be used to self-test any partially
entangled state of two qubits. The optimal measurements, however, are not
unique and form a one-parameter family of qubit measurements. The second
branch, which includes the well-known $I_{3322}$ functional, is studied
numerically. We identify the region in the parameter space where the quantum
value can be attained, with two-dimensional systems and characterise the state
and measurements attaining this value. Finally, we show that the set of
realisations introduced in reference [Phys. Rev. A 82, 022116 (2010)] to obtain
the maximal violation of the $I_{3322}$ inequality succeeds in approaching the
optimal value for a large subset of the functionals in this branch. In these
cases we analyse and discuss the main features of the optimal realisations.
