Rebuilding trust without set parameters, boundaries and tools... is probably a great theory. There needs to be incremental progress shown. And the first step I would recommend are filters on all computers in the house.
No corporate owned computer hooked up to a network would allow access to every website out there. Too much opportunity to waste time, sexual harrassment of images and downloading viruses that can compromise security. (Hmmm... sound like familiar objectives here?)
Your home computers need to be setup in the same way - and that includes other devices like tablets and cell phones.
Often on the forum, the K9 web protection filter is recommended. It's free and it's pretty good. It'll work on everything... except Android smart phones.
K9 Web Protection - Free Internet Filter and Parental Control Software | Free Internet Filtering and Parental Controls Software
I would search online for smartphone internet filters and see if they're there. If not, you may need to get a new phone that's compatible with other internet filters, or downgrade to a "dumb phone".
This is the first step for your situation. If he won't abide by this... then that may be your sign.
Of course there are ways around such filters. One very basic way is to reset your devices... or to reload your operating system onto the computers.
The first step is getting him to agree to having them installed.
How do you go about the conversation? More importantly, how can you have the conversation and make him a hero for doing it? (Now THAT takes some creativity!)
1. You're doing this for the safety of children who may use the devices at home. You don't want your children exposed to anything that they shouldn't be exposed to. Just like you want a clean and safe work environment, you want a clean and safe home environment. (Kids may play games on smartphones, right? Kids might watch a movie on the tablet? Kids go online on home computers.)
2. If you don't have kids, just talk about having a clean and safe home environment... just like major corporations require for their computers. The longer the filters are installed, his cravings will lessen over time.
3. Take your devices (all of them, not just his, but every device in the household) to the bishop or counselor to place the password. This way, it's not YOU doing it (and holding a kind of 'moral superiority' over him), but the Bishop or counselor, and they can see how willing he is to work at his marriage. It won't just be "telling them" about it... but you AND they can SEE his sincerity towards fixing the marriage.
If you have a large desktop computer, you can use a service like teamviewer to give the Bishop or counselor momentary control over the computer to set the password.
TeamViewer - Free Remote Control, Remote Access & Online Meetings
After this is done... CELEBRATE this step! It's a MAJOR step! Go out for dinner and spend some time together. He just admitted that he has a problem and took steps in front of others to fix it. Reward him for doing so.
(Yes, I know you don't feel like it right now... but if he doesn't feel rewarded, he may not work towards the next steps.)
These are just my thoughts.