gamma2 link function choice VGAM package of R

Question

I'm using the VGAM package in R with the gamma2 family function (two parameters: mu and shape). I'm hesitating in the choice of the link function. In fact, I have the two following fits:

fit1 <- vglm(y ~ x,
             family = gamma2(lmu ="identitylink", lshape="identitylink"),
             trace = TRUE,
             data = gdata,
             na.action =  "na.omit",
             maxit = 100)

fit2 <- vglm(y ~ x,
             family = gamma2(lmu ="loge", lshape="loge"),
             trace = TRUE,
             data = gdata,
             na.action =  "na.omit",
             maxit = 100)

The two fits had no problem converge, even if fit2 converges faster than fit1. Which one to choose, does it makes sense to compare AIC(fit1) and AIC(fit2) - of course on the exact same dataset - or do I have to rely on other criteria?