Answer: Yes, iodine does penetrate the skin. But it's not useful to use an iodine patch test to judge whole-body sufficiency.
The authors concluded (1):
"Our quantitative determinations prove that iodine which penetrates through the skin is removed only slowly from within this area into the body, thus forming an iodine depot in the skin for several days. In this prolonged retention of iodine within the skin, we see a favorable condition for a possible local prophylactic and therapeutic action."