The symmetric monoidal functor conjecture for black-boxing open Markov processes

Let $\bold{Mark}$ and $\bold{LinRel}$ be the symmetric monoidal bicategories whose objects are finite sets, whose morphisms are open Markov processes and linear relations respectively, and whose 2-morphisms are morphisms of open Markov processes and inclusions of linear relations respectively. For a finite set $S$, define $\blacksquare(S)={\mathbb R}^S \oplus {\mathbb R}^S$. For an open Markov process $S \stackrel{i}{\rightarrow} (X,H) \stackrel{o}{\leftarrow} T$, its black-boxing is the linear relation determined by

$$\blacksquare(S \stackrel{i}{\rightarrow} (X,H) \stackrel{o}{\leftarrow} T)=\{(i^*(v),I,o^*(v),O): H(v)+i_*(I)-o_*(O)=0\}\subseteq {\mathbb R}^S\oplus {\mathbb R}^S\oplus {\mathbb R}^T\oplus {\mathbb R}^T.$$

For any morphism of open Markov processes, black-boxing sends it to the inclusion $\blacksquare(X,H)\subseteq\blacksquare(X',H')$. Black-boxing functor conjecture. There exists a symmetric monoidal functor $\blacksquare\colon \bold{Mark}\to\bold{LinRel}$ with these assignments. This would show that the black-boxing construction is compatible with morphisms, composition, and the symmetric monoidal structures after passage from double categories to bicategories. The source presents this as the expected bicategorical analogue of its established double-functor result, and gives no resolution.