Something seems off with equation (5).

Just imagining Monte Carlo sampling it, the middle expectation will have a bunch of zeros due to the indicator function and the right expectation won’t.

I can make the middle expectation be as close to zero as I like by making the success threshold sufficiently high.

Ah yes you are right the rhs was meant to be proportional to the middle expectation (see the equation below), for equality the rhs needs to be multiplied by a normalization constant independent of theta. Note this doesn't affect the bounds as the constant is the same across equations. Will update the paper to incorporate.