Let $F$ 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 $*$-polynomial $f(y^+,z^+)=y^+z^+$ has a nonlinear image on the upper triangular matrix algebra $A_n=\mathrm { UT}_n(F)$ ($n\geq 4$) under the alternating elementary $\mathbb Z_2$-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 $n=4$ and $n=5$, we prove that the linear span of the image is the entire odd component $(A_n)_1$ and that the additive defect is strictly minimal, with Waring length $2$. Our proof is constructive: we compute the products $(A_n)_0^+(A_n)_1^+$ via the orbit structure of matrix units and provide explicit two-value decompositions for arbitrary odd upper triangular matrices.

In this paper, we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices $UT_n$. For positive integers $q\leq n$, we classify these images on $UT_{n}$ endowed with a particular elementary ${\mathbb {Z}}_{q}$-grading. As a consequence, we obtain the images of multilinear graded polynomials on $UT_{n}$ with the natural ${\mathbb {Z}}_{n}$-grading. We apply this classification in order to give a new condition for a multilinear polynomial in terms of graded identities so that to obtain the traceless matrices in its image on the full matrix algebra. We also describe the images of multilinear polynomials on the graded algebras $UT_{2}$ and $UT_{3}$, for arbitrary gradings. We finish the paper by proving a similar result for the graded Jordan algebra $UJ_{2}$, and also for $UJ_{3}$ endowed with the natural elementary ${\mathbb {Z}}_{3}$-grading.