Waring Length Two for the Superinvolutive L’vov–kaplansky Counterexample on Upper Triangular Matrices

Let F F upper F be an algebraically closed field of characteristic zero. Campedel, Fagundes and Ioppolo [‘Upper triangular matrices with superinvolution: identities and images of multilinear polynomials’, Bull. Braz. Math. Soc. (N.S.) 57 (2026), Article no. 27] recently established a qualitative break from the L’vov–Kaplansky conjecture by proving that the multilinear ∗ $*$ asterisk -polynomial f ( y + , z + ) = y + z + f(y⁺,z⁺)=y⁺z⁺ f left parenthesis y Superscript plus Baseline comma z Superscript plus Baseline right parenthesis equals y Superscript plus Baseline z Superscript plus has a nonlinear image on the upper triangular matrix algebra A n = UT n ( F ) Aₙ= UTₙ(F) upper A Subscript n Baseline equals upper U upper T Subscript n Baseline left parenthesis upper F right parenthesis ( n ≥ 4 n≥ 4 n greater than or equals 4 ) under the alternating elementary Z 2 Z₂ double struck upper Z 2 -grading and super-reflection superinvolution. We provide the strict quantitative counterpart to this phenomenon by determining the exact Waring length of the counterexample in the first nontrivial dimensions. For n = 4 $n=4$