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In Kiukas, Lahti, and Ylinen (2006, Journal of Mathematical Physics 47, 072104), the authors asked the following general question. When is a positive operator measure projection valued? A version of this question formulated in terms of operator moments was posed in Pietrzycki and Stochel (2021, Journal of Functional Analysis 280, 109001). Let T be a self-adjoint operator, and let F be a Borel semispectral measure on the real line with compact support. For which positive integers 
$p< q$ do the equalities 
$T^k =\int _{\mathbb {R}} x^k F(\mathrm {d\hspace {.1ex}} x)$, 
$k=p, q$, imply that F is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if 
$p$ is odd and 
$q$ is even, and negative otherwise. The case 
$(p,q)=(1,2)$ closely related to intrinsic noise operator was solved by several authors including Kruszyński and de Muynck, as well as Kiukas, Lahti, and Ylinen. The counterpart of the second problem concerning the multiplicativity of unital positive linear maps on 
$C^*$-algebras is also provided.
