I've used lather and a bottle to clean and condition strops, lather is what the old time barbers used. I just finished using lather on two that were not that dirty but dry enough to loose some of the draw. After drying for about 24 hours I applied some neats foot oil to moisturize, got the draw back.

The steps I use are;
Work up thick lather with tallow based soap, I use Williams, apply heavily with a brush, work in well (5 - 10 minutes) with a bottle, wipe clean with damp cloth (you'd be surprised how much grunge you'll get), wipe dry as possible with dry cloth then hang up to dry. Let dry for roughly 24 hours or until completely dry then apply neats foot oil and work in thoroughly.

Your strop doesn't look that bad and should shape up nicely, good luck!