Equality of rational blowdown depth and combinatorial depth

Let $W_{p,q}(\mathbf{n})$ denote the minimal symplectic filling of $(L(p,q),\xi_{can})$ corresponding to $\mathbf{n}=(n_1,\ldots,n_k)$ in the parameterizing set $\mathcal{Z}_k(\frac{p}{p-q})$. For $k\geq 3$, define $\operatorname{dpt}(\mathbf{n})$ to be the number of entries equal to $1$ in the interior of $\mathbf{n}$:

$$\operatorname{dpt}(\mathbf{n})=\left|\{i\mid 1<i<k\text{ and }n_i=1\}\right|.$$

The rational blowdown depth conjecture. The rational blowdown depth of $W_{p,q}(\mathbf{n})$ is equal to $\operatorname{dpt}(\mathbf{n})$.

The preceding result gives only the upper bound by $\operatorname{dpt}(\mathbf{n})$; the conjecture asserts that this bound is always sharp, so the minimum number of successive symplectic rational blowdowns needed to obtain the filling from the minimal resolution is determined by the interior entries equal to $1$.