The Strichartz estimates are a family of inequalities for the wave equation, which quantify its dispersive nature by bounding the "size" of a solution in terms of an appropriate "energy" at initial time. Typically these estimates admit *maximizers*, that is, solutions which attain the maximal size per unit energy, but such maximizers are rarely known. Most notably, Foschi (2004) found that solutions with a Poisson kernel initial data maximize the conformal Strichartz estimate on the Minkowski spacetime $\mathbb{R}^{1+3}$, and conjectured that the same should hold in arbitrary dimension $\mathbb{R}^{1+d}$. We discuss this conjecture via the Penrose conformal compactification, and we prove that it fails for $d=2, 4, 6,...$