Conjecture on simultaneous Waring identifiability for ternary forms of consecutive degrees

Let $d\geq 2$. Consider a general complex polynomial vector consisting of $d-1$ ternary forms of degree $d$ and one ternary form of degree $d+1$. A simultaneous Waring decomposition expresses all its components using the same linear forms, with scalar coefficients allowed separately in each component.

Simultaneous Waring identifiability conjecture. The polynomial vector admits a unique simultaneous Waring decomposition with

$$\frac{d^2+d+2}{2}$$

summands.

This conjecture proposes a family extending the known identifiable examples involving a ternary quadric and cubic, and two ternary cubics and a quartic. Its general validity was the classification problem motivating the paper and is not resolved by the supplied text.