A counterexample to a conjecture on simultaneous Waring identifiability

The new identifiable case appeared in \cite{AGMO}, together with the analysis on simultaneous identifiability of pairs of ternary forms recently developed in \cite{BG}, suggested the following conjecture towards a complete classification of all simultaneous Waring identifiable cases: for any $d \geq 2$, the general polynomial vectors consisting of $d-1$ ternary forms of degree $d$ and a ternary form of degree $d+1$, with rank $\frac{d^2+d+2}{2}$, are identifiable over $\mathbf{C}$. In this paper, by means of a computer-aided procedure inspired to the one described in \cite{AGMO}, we obtain that the case $d = 4$ contradicts the previous conjecture, admitting at least $36$ complex simultaneous Waring decompositions (of length $11$) instead of $1$.