I need advice - UPDATE

My husbands ex has contacted him 3 times since sept. He was in a very serious long term relationship with her and he still has feelings for her. He would never cheat on me or leave me. I know this in my bones and he has told me the same. He is very happy with our life and family. But he gets nostalgic when she contacts him and he responds but does not continue a conversation past a few messages. She contacted him today and he told me right away. He always does. He tells me he has nothing to hide and wants me to know. She didn’t say anything but sent him pictures when they were together. He responded with pictures of our new nephew. She knows that we live together and have a child. But she obviously doesn’t care. I’ve asked him to tell her we’re married and have another baby on the way and to ask her to stop contacting him but he won’t. He says it’s unnecessary. That she doesn’t need to know what’s going on in our life and a friendly hello once in a while isn’t a problem. How can I get him to stop her from contacting him? I’ve thought about doing it myself but I know she’ll go back to him and that it’ll cause a problem between us.

Update: After I posted this I called him and told him he needs to tell her we’re married. That’s it’s disrespectful to me and our family etc. A little while later he told me he told her. That we’re married and he’s happy and committed to his family. He told me again he loves his family and our life and isn’t going anywhere. I felt better after that but something in my gut made me snoop. I had blocked her number and the numbers of everyone associated with her a long time ago so she’s been contacting him through email. I have access to his computer so the previous emails I had seen and were exactly as he told me. However this one is not. He told me it was just a picture of him with an Instagram filter and told me which picture of our nephew he sent plus the conversation they had after. Well it’s not a picture it’s a video of both of them in his car. The picture of the nephew is entirely different than the one he told me. And that’s it. There are no emails after. I opened up iMessage to see if he unblocked her and they were texting but nothing there. So they either spoke on the phone bc I can’t see his calls on the computer, they texted through WhatsApp or Viber or he lied to me about coming clean to her just to pacify me.