This is a follow-up question on this. Let $A$ be a von Neumann algebra and $P$ be its projection lattice.

For $p,s,q \in P$, let us define $ p \perp q \mid s \iff ps^\perp q = 0$ where $s^\perp = 1-s$.

Let $P_0 \subset P$ be a complete sublattice of $P$. Is the following true over $P_0$ (i.e., for any $p,q,r,s \in P_0$): \begin{align*} r \perp p \mid (q \vee s), \quad r \perp q \mid (p \vee s) \implies r \perp (p\vee q) \mid s\;? \end{align*}