The length of the body diagonal of a rectangular prism is L. What is the largest possible volume enclosed by the prism?

So I said that the dimensions are x, y, and z. Using a Pythagorean theorem:

Using this to solve for each side:

Then substituting back into the equation for volume:

which simplifies to:

But then you can say:

And then substituting into the simplified volume equation:

I don't see any problems with my math, but the answer doesn't seem intuitively correct to me. Maybe it is, but it just doesn't feel right at all. Can anyone conclusively confirm or deny this answer?