As reported by @vzw_customer_support , @PamD_VZW on the other thread the workaround is to use the web site access for messaging and not the desktop app.  However, I opened the https://web.vma.vzw.com website for access to Verizon Messaging and it STILL DOES NOT SYNC my messages to my contact using a Galaxy Core Prime phone on MagicJack/Ymax mobile.  Their messages to me are sync'ed fine but mine NOT.  Both directions are sync'ed on my Droid Turbo 2 on the VZW account.   Also, my messages to an iPhone on VZW were NOT sync'ed before on the desktop app and they ARE STILL NOT SYNC'ed on the web site access.  SAME ISSUE ON YOUR WORKAROUND AS THE 'INCOMPATIBLE DESKTOP APP' FOR ME.   IT IS APPARENT THAT THE NON-SYNC'ING MESSAGES REACHED AND ARE AVAILABLE IN THE MESSAGING SERVER BUT FOR SOME REASON, NOT BEING FORWARDED TO THE DESKTOP APP NOR THE WEB SITE ACCESS.