On the equivalence of linear sets

Let $L$ be a linear set of pseudoregulus type in a line $\ell$ in $\Sigma^*=\mathrm{PG}(t-1,q^t)$, $t=5$ or $t>6$. We provide examples of $q$-order canonical subgeometries $\Sigma_1,\, \Sigma_2 \subset \Sigma^*$ such that there is a $(t-3)$-space $\Gamma \subset \Sigma^*\setminus (\Sigma_1 \cup \Sigma_2 \cup \ell)$ with the property that for $i=1,2$, $L$ is the projection of $\Sigma_i$ from center $\Gamma$ and there exists no collineation $\phi$ of $\Sigma^*$ such that $\Gamma^{\phi}=\Gamma$ and $\Sigma_1^{\phi}=\Sigma_2$. Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des. Codes Cryptogr. 56:89-104, 2010) states the existence of a collineation between the projecting configurations (each of them consisting of a center and a subgeometry), which give rise by means of projections to two linear sets. It follows from our examples that this condition is not necessary for the equivalence of two linear sets as stated there. We characterize the linear sets for which the condition above is actually necessary.