Equality of intrinsic and extrinsic reinforced spaces on the cube

Let $Q$ be the unit cube in $\mathbb{R}^n$ for $n\geq 2$. For $1\leq p<\infty$, $0<q<\infty$, and $s>\sigma_{p,q}$, let $F_{p,q}^{s,\operatorname{rinf}}(Q)$ denote the reinforced function space defined using a continuation outside the singular boundary strata, and let $F_{p,q}^{s,\operatorname{rinf}}(Q)^*$ denote the intrinsically reinforced space defined by boundary decay conditions from inside $Q$. Equality conjecture. One should have

$$F_{p,q}^{s,\operatorname{rinf}}(Q)=F_{p,q}^{s,\operatorname{rinf}}(Q)^*.$$

The intrinsic definition is known to contain the continuation-based space by restriction, so the conjecture asks whether every function satisfying the interior reinforce properties admits the corresponding global continuation.