A Confusing Logic Puzzle

Jane, Emily, and Mike are perfect logicians. One day, Jane said, "I'm thinking of four non-negative integers $a, b, x,$ and $y$ that obey the following conditions: $\begin{aligned} |a - x| &\geq 1\\ |a - x| &\geq \min(b, y)\\ |a - x| &\leq 1 + \min(b, y)\\ |b - y| &\leq 1. \end{aligned}$ Then Jane said, "I'm going to tell $a$ and $b$ to Emily and $x$ and $y$ to Mike."

Emily said, "I don't know $x$, and I wouldn't know it even if I knew whether $y$ was the same as $b$."
Mike said, "I don't know $a$, and I wouldn't know it even if I knew whether $b$ was the same as $y$."
Emily said, "I don't know $x$, and I wouldn't know it even if I knew whether $y$ was the same as $b$."
Mike said, "I don't know $a$, and I wouldn't know it even if I knew whether $b$ was the same as $y$."
Emily said, "I don't know $x$, and I wouldn't know it even if I knew whether $y$ was the same as $b$."
Mike said, "I don't know $a$, and I wouldn't know it even if I knew whether $b$ was the same as $y$."

Jane then interrupted, "Stop! You two could go on forever like that!"

Emily said, "I didn't know that."
Mike said, "I didn't know Emily didn't know that. If Emily had said she knew that, I wouldn't know whether Emily knew whether $x$ is greater than $15$. But now, I do."
Emily said, "Before Mike said that, I didn't know whether Mike knew which of $a$ and $x$ is bigger."

What is the maximum value of $a \times b+x \times y?$