It is well known that no deterministic two-party protocol can solve disjointness problem (DISJ) on $n$-bit inputs without sending $n+1$ bits in the worst case (see, e.g., the book by Kushilevitz and Nisan). For bounded-error randomized protocols, a lower bound of $\delta n$, for some constant $\delta$, has also been shown in a seminal paper by Razborov [Razborov92]. My question is:
What is the best known explicit value of $\delta$ currently (both upper and lower bounds)?
Also, is there any belief on the actual value of $\delta$?
[Razborov92] Alexander A. Razborov: On the Distributional Complexity of Disjointness. Theor. Comput. Sci. 106(2): 385-390 (1992) doi:10.1016/0304-3975(92)90260-M