Is the white hole a myth or not?

Well, currently, we don't have a shred of evidence that they exist.

White holes are hypothetical objects predicted by certain solutions to Einstein's theory of general relativity. To be specific, they come from the maximally extended Schwarzschild solution. We can deal with them a little mathematically, but nobody will really believe in objects like these unless we have experimental evidence for them.

Do we? I quote NASA here

White holes are VERY hypothetical.

and here

There is no observational evidence for white holes.

This supports this position:

Unlike black holes, there is no real astronomical evidence to suggest that white holes actually exist.

Also, the idea is put nicely here:

A white hole is only a concept for higher levels of thinking. No one has every observed one and no one probably ever will.

Why? This gives an interesting answer:

Once even the tiniest speck of dust enters the part of space-time which includes the black hole, the part which includes the white hole disappears. The universe has been around for a long time and so even if it did start with white holes, they would have all disappeared by now. (Emphasis mine)

Essentially, white holes are unstable. Even if a particle did try to enter one from a black hole via an Einstein-Rosen bridge, the bridge would collapse before the particle could cross, and the particle would hit the singularity.