Can I disable the UIPageViewController's page border gesture recognizers? And keep the swipe one?

Try looping through pageViewController.gestureRecognizers, disabling any that are tap recognizers.

Objective-C:

for (UIGestureRecognizer *recognizer in pageViewController.gestureRecognizers) {
    if ([recognizer isKindOfClass:[UITapGestureRecognizer class]]) {
        recognizer.enabled = NO;
    }
}

Swift:

for recognizer in pageViewController.gestureRecognizers {
    if recognizer is UITapGestureRecognizer {
        recognizer.isEnabled = false
    }
}