A sharpened energy-Strichartz inequality for the wave equation

We consider the sharp Strichartz estimate for the wave equation on ${\mathbb{R}}^{1+5}$ in the energy space, due to Bez and Rogers. We show that it can be refined by adding a term proportional to the distance from the set of maximisers, in the spirit of the classical sharpened Sobolev estimate of Bianchi and Egnell.