Representations of cyclotomic oriented Brauer categories

Let $A$ be the locally unital algebra associated to a cyclotomic oriented Brauer category over an arbitrary algebraically closed field $\Bbbk$ of characteristic $p\ge 0$. The category of locally finite dimensional representations of $A$ is used to give the tensor product categorification (in the general sense of Losev and Webster) for an integrable lowest weight with an integrable highest weight representation of the same level for the Lie algebra $\mathfrak g$, where $\mathfrak g$ is a direct sum of copies of $\mathfrak {sl}_\infty$ (resp., $\hat{\mathfrak {sl}}_p$ ) if $p=0$ (resp., $p>0$). Such a result was expected in [3] when $\Bbbk=\mathbb C$ and proved previously by Brundan in [2] when the level is $1$.