You speak of the 4 Ohio class boats converted to SSGN. You will notice that those had ALL their ballistic missiles removed and totally replaced with Tomahawks. Those 04 boats do not carry ANY nuclear payload in their current configuration. At the same time NONE of the remaining 14 Ohio class SSBN's carry ANY tomahawk missiles. The roles are deliberately and completed separated and this conversion was done AFTER the end of the Cold War as part of the START Treat reductions in force. The USN simply found a better use for the 4 hulls than scrapping them..but it was done with agreement from the Russians.There may have been such unwritten conventional rules between usa and ussr but i doubt it, as Usa had only SSBN s at that time, no diesel electric subs in operation and they still doesn't operate any diesel sub even now. You are saying that Usa doest have conventional warhead on their missiles in their SSBNS.... no way they have...they have used it in middle east...so your argument is i am afraid to say is baseless.
No SSBN carries any non-nuclear / conventional payload missiles simultaneously! NONE I'm not talking out my ass about these issues...I work in naval industry
The USN had studied a program called Prompt Global Strike which involved putting conventional payload ballistic missiles on their SSBNs to strike HVT's in the global war on terror. You can read about why that program was rightfully shelved