The reason McQueen wandered around the prison looking for a way to kill himself was not just for the sake of killing himself, there were plenty of ways he could have gone about it. The reason he didn't commit suicide immediately whenever an opportunity presented itself was because of his desire to not die alone. Also, the person who he chooses to die with him will have to be someone special/significant to him, and not some random person on the streets.
In fact, the reason he targets Ermes is because she took his memory disc by mistake and he forgot that he had stolen her money, so when she tried to dig up info on him and followed him, he mistook her for having feelings for him. That's why he tried for the lover's suicide with Ermes and not the tons of other people he comes in contact with on a daily basis. Also, having his memory disc removed effectively removed every other good memories he may have had which coupled with the nature of his Stand turned him into a cripplingly depressed person contemplating suicide with a special someone, when he finds her.
If TMQ is having CH as his opponent(read suicide partner), it means he already has strong feelings for him/her and is likely to have activated HtH on CH already and not have interacted with anyone else(most of whom will not interact with him and those who do will not leave strong enough impressions on him). TMQ will not have seen CH for 10 days or more, so he'll not commit suicide immediately(because he needs confirmation that CH is indeed going to die with him).
Suicide watch tactics will not work. If TMQ drowns himself, CH can't CPR him if he's choking or passed out himself. If he hangs himself, CH will also be choking while levitating and unable to properly move. If TMQ electrocutes himself, CH also gets incapacitated or dies. Tongue biting is the only thing CH can possibly save TMQ from as he can heal himself first and attend to TMQ, but as someone already said, TMQ goes for conventional suicide methods(tongue biting is not something anyone can do. Its something the military personnel or secret service people might do if they're captured. Its also not instantaneous and highly painful which is why most people would rather not do it.)
I'd say inconclusive. If CH stays out of sight and waits, nothing happens leading to inconclusive. If CH tries to kill, he dies.
