Strong contextuality conjecture for universally transversal code-switching protocols

Let $\mathcal{Q}_1$ and $\mathcal{Q}_2$ be subsystem stabilizer codes, and consider a fault-tolerant code-switching protocol between them. Let the set of transversal gates for each code be the gates implementable transversally on that code. Strong contextuality conjecture. If the union of the set of transversal gates for $\mathcal{Q}_1$ and that for $\mathcal{Q}_2$ forms a universal gate set, then the code-switching protocol must be strongly contextual in a partial closure. This is proposed as a no-go principle for noncontextual routes to transversal universality through code-switching; the supplied text gives no resolution, so the conjecture remains open.