Orthogonally additive polynomials on non-commutative $L^p$-spaces

Let $\mathcal{M}$ be a von Neumann algebra with a normal semifinite faithful trace $\tau$. We prove that every continuous $m$-homogeneous polynomial $P$ from $L^p(\mathcal{M},\tau )$, with $0<p<\mathrm{\infty }$, into each topological linear space $X$ with the property that $P(x+y)=P(x)+P(y)$ whenever $x$ and $y$ are mutually orthogonal positive elements of $L^p(\mathcal{M},\tau )$ can be represented in the form $P(x)=\mathrm{\Phi }(x^m)$ $(x\in L^p(\mathcal{M},\tau ))$ for some continuous linear map $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:L^{p\mathrm{/}m}(\mathcal{M},\tau )\to X$.