Gutin's uniqueness conjecture for maximal self-adjoint decompositions

Let $\mathcal{A}$ be a finite-dimensional $\ast$-algebra over $\mathbb{R}$, and let $H\in\mathcal{A}$ be self-adjoint, meaning $H^{\ast}=H$. Let $k\in\mathbb{N}$ be maximal among the integers for which there are nonzero self-adjoint pairwise orthogonal elements $P_1,\ldots,P_k\in\mathcal{A}$ satisfying

$$H=\sum_{i=1}^k P_iHP_i.$$

More explicitly, $P_i^{\ast}=P_i$, $P_iP_j=\delta_{ij}P_i$, and $P_i\neq 0$ for all $i,j\in[k]$. Consider another decomposition of the same maximal length,

$$H=\sum_{i=1}^k Q_iHQ_i,$$

with the analogous conditions on the $Q_i$. Gutin's conjecture. There exist a permutation $\sigma\in S_k$ and an element $U\in\mathcal{A}$ such that $U^{\ast}=U^{-1}$, $UHU^{\ast}=H$, and

$$UQ_iU^{\ast}=P_{\sigma(i)}\qquad\text{for all }i\in[k].$$

This conjecture proposes a broad uniqueness principle generalizing the spectral theorem: maximal decompositions of a self-adjoint element in a finite-dimensional real $\ast$-algebra should be related by a unitary element preserving that element, up to permutation. The paper states that certain cases are proved and presents applications to representing matrix decompositions through $\ast$-algebras and $\ast$-ideals; the general claim remains open.