Im trying to maximize the probability of a particular outcome occurring subject to a constraint. In particular

$$\max \prod_{i \leq n} 1 - (1 - x_i)^{y_i} \;\;\; \text{ s.t. } \;\;\; i \in \mathbb{N}^+,\; 0 \leq x_i \leq 1,\; x_1\cdots x_n = z, \forall n \in \mathbb{N}^+$$

where $y_i \in \mathbb{N}^+$ and $0 \leq z \leq 1$. The context really isn't important, I'm interested only in a solution to this problem. I've been able to find and prove a solution for the minimum, but I haven't been able to for the max.

I highly suspect that the maximum is at $x_1 = \cdots = x_n = z^{(1/n)}$ ($y_i$ fixed for all $i$), but I have not been able to come close to proving this. I'm looking for a proof that either the solution that I have proposed is correct, or incorrect. I'm not necessarily looking for a solution, but it would be welcome. I've been trying to prove this for quite some time and haven't had any luck (proving it false or true). Any advice or suggestions would be greatly appreciated.

Note (read all first): I've posted this on math exchange, but despite the number of views, I haven't received any responses. I'm reposting here (shouldn't do this, I know, but in retrospect I think this question is more relevant here) because I'm beginning to wonder if this problem is in fact much more difficult than I originally thought. It seems to me that someone would have looked at a problem similar to this from the research community since this problem, at least to me, seems relatively elementary despite its potential usefulness when calculating outcome probabilities. Barring a solution, is anyone aware of any references that I could take a look at that might lead me to a proof or counter proof?

In addition I have included algebraic geometry as a tag because one of the approaches I have looked at is a reduction of this problem by looking at it as the maximization of a $n$ hyper rectangle. That is to say given a hyper rectangle with $n$ dimensional volume $x_1\cdot...\cdot x_n = z$, what side lengths will give the largest $n$ dimensional volume if we set each side length to be $1 - (1 - x_i)^{y_i}$ for fixed $y_i$. From this perspective, I would expect the greatest $n$ dimensional volume increase (and thus greatest volume) would occur when all sides ($x_i$) have the same length. I don't have much of a background in geometry though so I haven't gotten far on this.

Edit.

I mentioned that I was able to prove what the minimum was. My proof was incorrect. Doesn't change this question, but I wanted to make sure the problem description was as accurate as possible.