Dirichlet infinite-linking convergence conjecture

Let $M$ be the material manifold and let $\mathbb{M}_0=[0,\tau]\times M$ be the spacetime manifold with Dirichlet boundary conditions. For each $a\geq 0$, let $H_{\mathrm{dir},a}$ and $S_{\mathrm{dir},a}$ denote the geometric and functional minimisation constants on $\mathbb{M}_0$, and let $h_{\mathrm{dir}}^D$ and $s_{\mathrm{dir}}^D$ denote the corresponding material constants on $M$. Let $F_a:\mathbb{M}_0\to\mathbb{R}$ minimise $S_{\mathrm{dir},a}$, let $f:M\to\mathbb{R}$ minimise $s_{\mathrm{dir}}^D$, let $\mathbb{A}_a$ minimise $H_{\mathrm{dir},a}$, and let $A$ minimise $h_{\mathrm{dir}}^D$.

Dirichlet infinite-linking convergence conjecture. The constants and minimisers satisfy

$$\lim_{a\to\infty} S_{\mathrm{dir},a}=s_{\mathrm{dir}}^D\quad\text{and}\quad\lim_{a\to\infty} H_{\mathrm{dir},a}=h_{\mathrm{dir}}^D.$$

Moreover,

$$\lim_{a\to\infty}F_a(t,x)=f(x)$$

for all $t\in[0,\tau]$ and $x\in M$, and

$$\lim_{a\to\infty}\mathbb{A}_a=[0,\tau]\times A.$$

This is the Dirichlet-boundary analogue of the convergence conjecture for the mixed boundary condition setting: infinitely tight linking should recover the corresponding material minimisers.