Embeddable partial Hadamard matrices related to the projective planes

The existence of a projective plane of order $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$, where $p$ is a prime power, is shown to be equivalent to the existence of a balancedly multi-splittable embeddable $p^2\times p(p+1)$ partial Hadamard matrix.