Embedding conjecture for three-dimensional symmetric designs from regular Hadamard matrices

Let $H$) be a regular Hadamard matrix of order $v^2$. A three-dimensional embedding conjecture asserts that there exists a $\mathcal{C}^3_3(v^6,v^8(v^4-1)/2,v^8(v^4-2)/4)$-cube containing a $\mathcal{C}^3_3(v^3,v^4(v^2-1)/2,v^4(v^2-2)/4)$-subcube.

Embedding conjecture. If $H$ is a regular Hadamard matrix of order $v^2$, then such a containing cube and subcube exist.

The existence would follow from the cited theorem if $H$ could be embedded in a regular Hadamard matrix of order $v^4$; the authors note that they are unaware of results on embedding regular Hadamard matrices.