High-rank ternary forms of even degree

We exhibit, for each even degree, a ternary form of rank strictly greater than the maximum rank of monomials. Together with an earlier result in the odd case, this gives the lower bound $$\operatorname{r_{max}}(3,d)\ge\left\lfloor\frac{d^2+2d+5}4\right\rfloor$$ for $d\ge 2$, where $\operatorname{r_{max}}(n,d)$ denotes the maximum rank of degree $d$ forms in $n$ variables with coefficients in an algebraically closed field of characteristic zero.