Is $(2^a - 3^b)$ mutually prime with $(2^a - ((2^a)^{-1} \mod 3^b ) - ((3^b)^{-1} \mod 2^a))$?

Question

Is $(2^a - 3^b)$ mutually prime with $(2^a - {{[2^a]}^{-1}}\mod 3^b + [3^b]^{-1}\mod 2^a)$, provided that $a\geq b$ and $b>0$?

By $[2^a]^{-1}\mod 3^b$ I mean "the multiplicative inverse of $2^a\mod3^b$".