It might we worthwhile to talk about this density instead $$\frac{\ln(|\{K: K \text{ is monogenic} \Delta(K)< x\}|)}{\ln(|\{K: \Delta(K) < x\}|)}$$ which is expected to be $$\frac{1}{2}+\frac{1}{n}.$$ Combining the results in Bhargava, Shankar, Wang 's paper ``Polynomials with squarefree discriminants I", with Malle's Conjecture we can conclude that the liminf of the above is greater than or equal to $$\frac{1}{2}+\frac{1}{n}.$$
In the same paper, it is also shown that the second desity is infact $$\frac{6}{\pi^2}$$