# Local vol vs stochastic vol in the context of American digital options

I have two models of some spot. One is under local vol and the other is under stoch vol. Both are calibrated to the prevailing vanilla prices. I then consider the option that pays $$1$$ if the spot price exceeds a certain level before expiry, $$T$$. If there is any payment to be made, this happens at the time the barrier is reached.

$$S$$ denoting the spot price process, we can define

$$\tau = \inf\{t \geq 0: S_t = A\}$$

with $$A$$ being the barrier and $$S_0 < A$$. The price of the option is then given by $$E[e^{-r\tau} \mathbb{1}_{\tau \leq T}]$$

with $$r$$ being the interest rate. Under which model (local or stoch) is this price higher?

Does the answer change if the payout occurs at $$T$$? Would it change if $$S_0 > A$$?

I am looking for an answer that is not entirely intuitive but one that justifies itself at least with some hand-wavy calculations.