The asymptotic leading term for maximum rank of ternary forms of a given degree

Let rmax(n,d) be the maximum Waring rank for the set of *all* homogeneous polynomials of degree d>0 in n indeterminates with coefficients in an algebraically closed field of characteristic zero. To our knowledge, when n,d >= 3, the value of rmax(n,d) is known only for (n,d)=(3,3),(3,4),(3,5),(4,3). We prove that rmax(3,d)=d^2/4+O(d) as a consequence of the upper bound on rmax(3,d) given by the floor of (d^2+6d+1)/4.