The characterization of 1-trivial motions for size-k psd factorizations

Let $\mathcal{M}_{\binom{k+1}{2},k}^{p \times q}$ denote the matrices admitting size-$k$ positive semidefinite factorizations, and let $L_{(I,I)}\mathcal{M}_{\operatorname{GL}(k)}$ denote the tangent space at $(I,I)$ to the orbit induced by the action of $\operatorname{GL}(k)$. A size-$k$ psd factorization has a $1$-trivial motion when its infinitesimal motion is induced by this group action.

The 1-trivial-motion conjecture. The set of $1$-trivial motions of size-$k$ psd factorizations of matrices in $\mathcal{M}_{\binom{k+1}{2},k}^{p \times q}$ is equal to $L_{(I,I)}\mathcal{M}_{\operatorname{GL}(k)}$.

The paper explicitly verifies this description for a size-$2$ example and proposes it for arbitrary $k$; no resolution is supplied.