Exact maximum Waring rank of ternary sextic forms

Determine the exact maximum Waring rank among ternary homogeneous polynomials of degree 6; that is, determine the largest integer s such that there exists a ternary sextic that cannot be expressed as a sum of fewer than s sixth powers of linear forms.

Background

The Waring rank of a ternary sextic is the smallest number of sixth powers of linear forms needed to represent the polynomial. For a general ternary sextic, dimension-counting bounds imply at least 10 summands are needed, and indeed 10 suffice. However, the worst-case (maximum) rank exceeds this generic value.

Existing results provide a lower bound of at least 12 for the maximum rank, but the precise maximum is not currently known.