  For the purpose of proving the strong openness conjecture of multiplier ideal
sheaves, Jonsson-Musta\c{t}\u{a} posed an enhanced conjecture and proved the
two-dimensional case, which says that: the Lebesgue measure of the set
$\big\{c_o^F(\psi)\psi-\log|F|<\log r\big\}$ divided by $r^2$ has a uniform
positive lower bound independent of $r$, for a plurisubharmonic function $\psi$
and a holomorphic function $F$ near the origin $o$. Jonsson-Musta\c{t}\u{a}'s
conjecture was proved by Guan-Zhou depending on the truth of the strong
openness conjecture. However, it is still a question whether one can prove
Jonsson-Musta\c{t}\u{a}'s conjecture without using the strong openness
property, and obtain a sharp effectiveness result for this conjecture.
  In this article, we use an $L^2$ method with the weight functions
$\psi-\log|F|$ and firstly consider a module at at a boundary point of the
sublevel sets of a plurisubharmonic function. By studying the minimal $L^{2}$
integrals on the sublevel sets of a plurisubharmonic function with respect to
the module at the boundary point, we establish a concavity property of the
minimal $L^{2}$ integrals. As applications, we obtain a sharp effectiveness
result related to Jonsson-Musta\c{t}\u{a}'s conjecture, which completes the
approach from the conjecture to the strong openness property. We also obtain a
strong openness property of the module and a lower semi-continuity property
with respect to the module.
