Consequences of the simple product conjecture for compatible families

Let $\mathscr B(\infty)$ be the crystal indexing the dual canonical basis. A family is called compatible or strongly compatible as in the source, and $b^{[m]}$ and $S_m(b)$ denote the associated crystal elements. Write $b_i\bot b_j$ for the source's compatibility relation. Simple product conjecture. (1) If $b_1,\ldots,b_m\in\mathscr B(\infty)$ satisfy $b_i\bot b_j$ for every $1\leq i<j\leq m$, then they form a compatible family. (2) Every real element $b\in\mathscr B(\infty)$ is strongly real and satisfies $b^{[m]}=S_m(b)$ for any $m$. (3) Every compatible family of real elements is strongly compatible. These assertions are presented as consequences expected from the open orbit and quantized categorification conjectures; the source gives no resolution status.