Non-equivalence of strong and weak Morrey quasiconvexity

Throughout, let $Q$ be the unit cube in $\mathbb R^n$. A measurable function $f:\mathbb R^{nm}\to\mathbb R$ is strong Morrey quasiconvex if, for every $\epsilon>0$, $A\in\mathbb R^{nm}$, and $K>0$, there exists $\delta=\delta(\epsilon,K,A)>0$ such that every $\varphi\in W^{1,\infty}(Q,\mathbb R^m)$ satisfying

$$\lVert D\varphi\rVert_{L^{\infty}}\le K,\qquad \max_{x\in\partial Q}|\varphi(x)|\le\delta$$

also satisfies

$$f(A)\le\operatorname{ess\,sup}_{x\in Q}f(A+D\varphi(x))+\epsilon.$$

It is weak Morrey quasiconvex if, for every $A\in\mathbb R^{nm}$ and every $\varphi\in W^{1,\infty}_0(Q,\mathbb R^m)$,

$$f(A)\le\operatorname{ess\,sup}_{x\in Q}f(A+D\varphi(x)).$$

Non-equivalence conjecture. If $m,n>1$ and $f$ is lower semicontinuous, then strong Morrey quasiconvexity and weak Morrey quasiconvexity are not equivalent.