I've used this gator tape on a lot of old wood steps. Sometimes it works great but freezes while others not so much. For the best results you should clean your steps really well and dry them beforehand. It can also be good to sand the finish to get to the wood, otherwise the glue won't stick. Additionally, being in the Northwest with a lot of humidity in the air does not promote adhesive adhesion. They work great until the glue starts to fit. More recently I've also started sewing the edges and in some cases this has helped.
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