I would definately agree with your theory. However, in practice, I have not seen this to work. I have always had very low SHBG, and my diet has always been VERY high in fiber. We're talking 4-5 cups of vegetables per day, plus a little oatmeal, and some fiber powder -- I aim for 40-60g of fiber per day.
I can say confidently that after being on a high (40-60g) fiber diet, I still have very low SHBG (1 unit above lowest end of normal range.)