Existence of DODDs for arbitrary matrices in sufficiently large dimensions

Let $\mathbf{X}\in\mathbb{R}^{m\times n}$, and for an integer $d\geq m,n$ let its 0-inflation be the $d\times d$ matrix obtained by placing $\mathbf{X}$ in the top-left corner and filling the remaining entries with zeros. A DODD is a decomposition involving diagonal matrices and an orthogonal matrix; write the corresponding rectangular block as $\overline{\mathbf{Q}}$. Equivalently, let $\mathbf{C}\in\mathbb{R}^{m\times d}$ have orthonormal rows and let $\mathbf{F}\in\mathbb{R}^{d\times n}$ have orthonormal columns.

DODD existence conjecture. The $d\times d$ 0-inflation of any matrix $\mathbf{X}\in\mathbb{R}^{m\times n}$ admits a DODD for some sufficiently large $d\geq m,n$. Thus every such matrix has a decomposition

$$\mathbf{X}=\boldsymbol\Lambda\overline{\mathbf{Q}}\mathbf{M},$$

where $\boldsymbol\Lambda\in\mathbb{R}^{m\times m}$ and $\mathbf{M}\in\mathbb{R}^{n\times n}$ are diagonal, and $\overline{\mathbf{Q}}\in\mathbb{R}^{m\times n}$ is the top-left block of an orthogonal matrix $\mathbf{Q}\in\mathbb{R}^{d\times d}$. Equivalently,

$$\mathbf{X}=\boldsymbol\Lambda\mathbf{C}^{\top}\mathbf{F}\mathbf{M}.$$

The conjecture would establish existence of such orthogonal-diagonal decompositions for every rectangular matrix after sufficiently large zero inflation. The source supports the conjecture only with numerical experiments, so its resolution remains open.