Consider Bernoulli bond percolation a locally finite, connected graph G and
let pcutβ be the threshold corresponding to a "first-moment
method" lower bound. Kahn (\textit{Electron.\ Comm.\ Probab.\ Volume 8,
184-187.} (2003)) constructed a counter-example to Lyons' conjecture of
pcutβ=pcβ and proposed a modification. Here we give a positive
answer to Kahn's modified question. The key observation is that in Kahn's
modification, the new expectation quantity also appears in the differential
inequality of one-arm events. This links the question to a lemma of
Duminil-Copin and Tassion (\textit{Comm. Math. Phys. Volume 343, 725-745.}
(2016)). We also study some applications for Bernoulli percolation on periodic
trees