  We focus on the $\mathcal{H}_2$-gain-scheduling synthesis problem for
time-varying parametric scheduling blocks with scalings. Recently, we have
presented a solution of this problem for $D$- and positive real scalings by
guaranteeing finiteness of the $\mathcal{H}_2$-norm for the closed-loop system
with suitable linear fractional plant and controller representations. In order
to reduce conservatism, we extend these methods to full block scalings by
designing a triangular scheduling function and by introducing a new lifting
technique for gain-scheduled synthesis that enables convexification.
