Change all pixels below 50% gray into 50% gray

Question

In a grayscale image, how can I lighten all pixels that are darker than 50% gray to 50% gray without changing any of the other pixels?

For example, this image:

should become this image:

I have tried using both the levels tool and the histogram. In the levels tool, I only managed to shift all pixel values, including those that are lighter than 50% gray:

In the historgram, I'm unable to draw an edge and the curve always curves around the midpoint:

Is there an easier solution?

Answer 1

Use Levels rather than curves to eliminate the splining curves generates.
Set input & output levels to 127 - 255 instead of 0 - 255.

Result

Answer 2

A layer in Lighten blending mode, filled with 50% gray. Lighten layer takes a look at the color below and blend color, and it keeps whichever one of the two is the lightest.