  For every $d \geq 2$, we present a generalization of the CHSH inequality with
the property that maximal violation self-tests the maximally entangled state of
local dimension $d$. This is the first example of a family of inequalities with
this property. Moreover, we provide a conjecture for a family of inequalities
generalizing the tilted CHSH inequalities, and we conjecture that for each pure
bipartite entangled state there is an inequality in the family whose maximal
violation self-tests it. All of these inequalities are inspired by the
self-testing correlations of [Nat. Comm. 8, 15485 (2017)].
