The transverse-distance conjecture for the $E_n$ D3-brane probe

Let $X$ be a local Calabi–Yau threefold containing a contractible surface $V_2\sim dP_n$, and let a D3-brane probe be placed near the associated $E_n$ 7-branes. Write $\operatorname{vol}(V_2)$ for the volume of $V_2$ in $X$.

Transverse-distance conjecture. The transverse distance between the D3-brane probe and the $E_n$ 7-branes is proportional to $\operatorname{vol}(V_2)$ in the local Calabi–Yau threefold $X$.

The conjecture is motivated by the proposed realization of the four-dimensional Minahan–Nemeschansky $E_8$ theory when $V_2$ shrinks to a point and wrapped M2- and M5-brane modes become massless; no proof or disproof is given.