Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian

Denoting the Hausdorff dimension of the Fibonacci Hamiltonian with coupling $\lambda$ by $\mathrm{HD}_\lambda$, we prove that for all but countably many $\lambda$, the Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian with coupling $\lambda$ is $\min\{2\mathrm{HD}_\lambda, 1\}$. Our proof relies on the dynamics of the Fibonacci trace map in combination with the recent result of M. Hochman and P. Shmerkin on the Hausdorff dimension of sums of Cantor sets which are attractors of regular iterated function systems (Local entropy averages and projections of fractal measures, Ann. Math. 175 (2012), 1001--1059).