The conjecture on strong convergence of the renormalized sequence

Let $u_k$ be the sequence considered in the blow-up argument, let $N_\infty$ be its limiting frequency, and define the renormalized sequence $v_k$ so that $v_k\to v$ weakly in $W^{1,2}(B_1)$. Here $B_1\setminus B_{1/2}$ is the annulus between the unit ball and the ball of radius $1/2$.

Strong-convergence conjecture. The renormalized sequence converges strongly:

$$v_k\longrightarrow v\quad\text{strongly in }W^{1,2}(B_1\setminus B_{1/2}).$$

Strong convergence would rule out loss of Dirichlet energy in the limiting blow-up argument and would support passing the relevant PDE information to the limit. The source explicitly notes that strong convergence is not clear and gives no resolution, so this conjecture remains open.