Interesting Proability Question

Just got an interesting probability question:

Suppose n points spread out in a 2D surface. Let's say the coordinates of those points are (x_i, y_i), i=1, 2, ..., n. Denote x_L, x_U the 25% and 75% quantile for (x_1, ..., x_n), and y_L, y_U the 25% and 75% quantile for (y_1, ..., y_n). p is the proportional of points lie in the small rectangle [x_L, y_L] *［x_U, y_U], then what is the range of p? (Assuming n is large)

It seems to me that if x and y are independent, then p is 1/4; but if x and y are perfectly correlated, say x=y, then p is 1/2. My guess is that 1/4 <= p <= 1/2, but anyone can verify this?