To simplify things, let's begin by considering a one touch option (that pays cash if the barrier is touched) rather than the double no touch option of the question. Also, let's assume interest rate and dividend yield are zero.

And before starting, we can remember that we are given a volatility surface at the start. Then both local and stochastic volatility models will value vanillas and digitals the same.

As [dm63][1] pointed out, an approximate hedge for the one touch is a digital with strike equal to the barrier level, and double the notional. At the moment the barrier touches, the digital is worth around 50% of its notional, so 100% of the one touch notional. As soon as the barrier touches, the digital hedge must be unwound.

In practice, this is not a perfect hedge for two reasons. Firstly, in a log-normal world, an at-the-money digital is not worth exactly 50%. That is easily dealt with by adding a vanilla option to the hedge that exactly corrects for this.

The second and more interesting reason is that a digital option has a correction to its Black-Scholes value equal to minus its Black-Scholes vega, multiplied by the derivative of the implied volatility (at the money) with respect to strike (hereafter, the "skew").

The upshot is that the value of the one touch at touch time can be replicated from a digital and a vanilla (whose values are independent of the model) plus a constant multiplied by the expected value of the skew at touch time.

In a stochastic volatility model, the smile tends to float when spot moves, meaning the at-the-money skew stays constant. In a local volatility model, the smile tends to stick in strike space when spot moves. As the smile is convex and at-the-money is near the bottom at the outset, this means the skew increases in magnitude. If one is careful with the sign of the factor multiplying the skew term, this gives the result.

It's important to be aware this is not a mathematical proof, and I have heard of claimed counter examples. For all practical cases though, it seems to be true.

Full details of the above calculation, and of the assumptions made are in chapter 9 of the book [Smile Pricing Explained][2].

Returning to the specific question, we have one touch plus no touch equals cash. Then as a one touch is higher in local vol, a no touch is lower in local vol and higher in stochastic vol. Deterministic interest rates and dividend yield can be included back in fairly easily.  Finally, we are asked to take on trust that the model behaviour of a double no touch is similar to a no touch. It is, and in fact, it is much more strongly model dependent and a single no touch. Even on a models FX volatility surface, a double no touch price can vary by a factor of two according to the model.

Finally it is worth mentioning that both volatility swaps and forward volatility agreements (forward starting vanilla options) are valued higher in local volatility than stochastic volatility. For the volatility swap, there is a beautiful proof (due to Dupire) in the very special case of an "instantaneous" volatility swap.

  [1]: https://quant.stackexchange.com/users/18388/dm63
  [2]: https://www.amazon.co.uk/Smile-Pricing-Explained-Financial-Engineering/dp/1137335718