Getting PFA + GCH above $\omega$

The proper forcing axiom is known to be indestructible by ${<}\aleph_2$-directed closed forcing, and since the forcing of the GCH for uncountable cardinals admits this degree of closure (iteratively add a Cohen set to the successor cardinals that are still there), it follows that one can simply force the GCH above $\aleph_1$ while preserving the PFA.

In Koenig and Yoshinobu's paper, they prove (Theorem 6.1) that $\sf PFA$ is preserved under $\omega_2$-closed forcings, and this should give the wanted result, as described by Joel.

Bernhard König and Yasuo Yoshinobu, Fragments of Martin’s maximum in generic extensions, MLQ Math. Log. Q. 50 (2004), no. 3, 297--302. MR 2050172