I've been thinking of this problem but I just can't complete a proof, here's a possible route (hopefully) towards a solution: By a simple continuity argument there is an such that on we have that does tend to zero. Now assume the following is true
Claim: If on the line with then on
then by an argument identical to the first on so the set on which has to be . I'm having a little trouble wih the claim though (particularly estimating in the verticla boundary of the set for some ; this is enough by the MMP).
Sorry I can't be of more help, if you find a proof please post it here.