I had the image rescanned and the lines are gone.
That's an interesting technique I hadn't heard of before - I'm going to try it anyway and learn something new.
Gave it a try, using some common techniques like gradient maps and luminosity fills. While those techniques even out the tones, they also destroy the film grain.
What looks best to me is a frequency separation technique. If you don't know "frequency separation", Google it. There are many samples and tutorials out there.
Frequency separation creates two different layers, one containing the detail only and one containing the tone only. It is used most often to clone and heal on the detail layer without touching the tone layer. But if you blur the tone layer, the result is an evening out of tones without losing detail, like the film grain.
In your sample, bluring the tone layer bleeds the darks and lights together, thus reducing the "line pattern". How much reduction depends on the amount of blur.