Sharpness conjecture for the stabilisation–singular distance inequality

Let $X$ be a smooth, compact, orientable $4$-manifold, and let $\Sigma$ and $\Sigma'$ be smooth, properly embedded, orientable, regularly homotopic surfaces in $X$. For surfaces $\Sigma$ and $\Sigma'$, write $d_{\operatorname{st}}(\Sigma,\Sigma')$ for their stabilisation distance and $d_{\operatorname{sing}}(\Sigma,\Sigma')$ for their singular distance.

Sharpness conjecture. There exist $X$, $\Sigma$, and $\Sigma'$ as above such that

$$d_{\operatorname{st}}\left(\Sigma,\Sigma'\right)=d_{\operatorname{sing}}\left(\Sigma,\Sigma'\right)+1.$$

The paper proves the general upper bound $d_{\operatorname{st}}(\Sigma,\Sigma')\leq d_{\operatorname{sing}}(\Sigma,\Sigma')+1$ and asks whether the additional $+1$ is essential. This conjecture asserts that the bound is sharp for some pair of regularly homotopic surfaces.