Visual Voicemails reappear after being deleted - iPhone 5S (7.0.4)

I have an iPhone 5S, 64GB, iOS 7.0.4 which has had a problem which appeared a couple of software updates back and is still broken after this latest 7.0.4 software update. I power-cycle my iPhone each day.

EVERYTIME I delete all the Visual Voicemails (whether I delete them one-by-one, after first listening to them or if I delete them using the Voicemail -> "Edit" -> "Swipe-Delete" method) they will reappear as Voicemails, appearing just as they did before I erased them.

I can even delete the deleted Voicemalis using the "Clear All" from the "Deleted Messages" folder in the Voicemail section and they will still reappear (Gone, but not gone?).

The voicemails may stay deleted for a little while, but they will reappear. One "trigger" that I have found which will cause the voicemails to reappear is an incoming phone call.

One consulation is that if you just keep deleting them, fewer & fewer of them will reappear.

I set up all my business appointments using my iPhone, so you can imagine how much fun this software flaw has become. Especially when you start getting those warnings that your voicemail space is almost full!

I called and opened an Apple Case on this item today when I failed to find any mention of it on any of the support boards (I do admit though, I'm so tired of this problem that I didn't look hard & heavy, I just want it fixed).

The first representative I contacted with Apple put me on hold many times and kept coming back to apoligize and then would go away again searching for solutions/answers/suggestions. None were offered. Not even one "Try this..." The final result of all those holds during the first call was that they didn't have a solution, didn't want my phone returned for repair (I bought AppleCare+), and wanted me to "just wait and watch it for a while" (watch what, how?). I asked if anyone else was reporting the same problem. The reply was that "my screen doesn't show that information."

After I discovered that an incoming call would trigger the return of the voicemails, I decided to call Apple back, to add that knowledge to the trouble ticket.

This second contact with Apple also resulted in many holds and excuses. I again asked if anyone else had reported the same problem. This time I was put on hold and then the agent told me to hold, that her supervisor wanted to speak with me. She returned to the call (no supervisor) and told me that there had been other reports of the same condition, but "fixes" were driven by importance and perhaps I wanted to report this condition on the iPhone support forum, as Apple developers did read the forums (my take is that it's the same at any tech company, including the one I retired from, customer embarassment can drive the priority of trouble resolutions and the testing and deployment of resolutions).

I guess I have a "low runner problem" (even though it's driving me nuts and making me cuss my 5S). I've been an Apple boy since 1977 (I was the first person to purchase an Apple ][ in Baton Rouge, Louisiana) and progressed through Apple IIe, Apple GS, then finally over to many, many models of MacsLisa, Quadra, the Mac 128K, the Mac 512K, Mac SE, LC, Performas, Power Macs (the names and models blurr, after a while), then as each new Mac Pro came out I'd migrate through those. Of course various iPods and then the purchase of my first iPhone and my iPad 4 gets mixed in with the rest of the pack... I'm pro Apple, make no mistake about it.

So I beg the folks out there that have this same or simular problem with their iPhones' visual voicemails reappearing after you delete them, and you're as tired of this as I am, please open an Apple Case on the issue.

I rest my case. Thank you for the time to read this posting...