RYBS has a well known shita about this. RYBS says that every minhag is either based on an existing din or a kiyum of an existing din (just like kol d'tikun rabbanan k'ein d'oraysa tiken). Therefore, he says that the minhag of aveilus that Ashkenazim have from the 17th of Tamuz must be based on an existing din, namely the 12 month Aveilus one has for a parent. Based on this he said that someone who shaved every day would be allowed to shave during the 3 weeks just like he is allowed to shave during the 12 months. However, the 9 days is like the aveilus of sheloshim which has no heter to shave and therefore it would be prohibited to shave during the 9 days even if you shave very day.
I personally do not shave during the 3 weeks for a very simple reason. The minhag in israel in general and specifically where I live is not to to shave and I have no compelling reason to shave, no one cares where I work whether I shave or not. This way I look different during the 3 weeks and it is an expression of the aveilus.