  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 $.
