The projection Unification Conjecture for mutual free information

Let $p$ and $q$ be projections in a $\mathrm{II}_1$-factor. The projection free entropy is denoted by $\chi_{\mathrm{proj}}$.

Projection Unification Conjecture. One has

$$i^\ast(W^\ast(p):W^\ast(q))=-\chi_{\mathrm{proj}}(p,q)+\chi_{\mathrm{proj}}(p)+\chi_{\mathrm{proj}}(q)=-\chi_{\mathrm{proj}}(p,q).$$

This is proposed as a natural extension of the Unification Conjecture from self-adjoint operator tuples to projections. The source presents it as a conjecture and gives no evidence of a resolution, so its status is open.