The internal stabilization conjecture for exotically knotted surfaces

Let $B^{4}$ be the smooth $4$-ball. For an exotically knotted pair of closed surfaces $(\Sigma_{1},\Sigma_{2})$ in $B^{4}$, let $d(\Sigma_{1},\Sigma_{2})$ denote the least nonnegative integer $k$ such that $\Sigma_{1}\#^{k}T^{2}$ and $\Sigma_{2}\#^{k}T^{2}$ are smoothly isotopic, where $T^{2}$ is an unknotted torus and $\#^{k}T^{2}$ denotes $k$ internal stabilizations. Internal stabilization conjecture. If $(\Sigma_{1},\Sigma_{2})$ is an exotically knotted pair of closed surfaces in $B^{4}$, then

$$d(\Sigma_{1},\Sigma_{2})=1.$$

Many families of exotically knotted surfaces are known to have stabilization distance one, but whether every such pair becomes smoothly isotopic after exactly one internal stabilization remains open.