Would You Caulk This Shower Tub?

Below is a photo of the shower tub in my new trailer. When I showered the first few times I notice some water leaking in the hall on the other side of the shower wall. Not a lot, but it was there. I realized that what was happening was that I unintentionally pointed the hand held shower at the edge of the tub where it meets the shower wall and some water seeped through to the other side. Rookie mistake, I know. I've been careful to avoid that since then and I haven't seen any more water leaking near the tub.

So I had the idea that I should caulk the edge of the tub and that would prevent any more leaks. I went to Walmart and bought some caulk, but I was cleaning the bathroom yesterday and I thought about how every bath I've ever owned or rented had caulk at the edge and it got the icky black or red mold that I could never completely clean out of the caulk. (I think 'icky' is the scientific term. I remember there was a TV show for kids in Dallas 50 years ago and the master of ceremonies was named Icky Twerp. He wore a cowboy hat that was much too small. I miss ol' Icky.)

So I have pretty much decided I'm not going to caulk this tub. I can take the caulk back to Walmart for a refund. Save your receipts. Do you agree? I think that may be why Jayco didn't caulk the tub in the first place. That and to save a few dollars.