They physically can't get too close else they are in range of shore based Iranian missiles, and having one of their precious ships sunk would force them to make a decision with no good options for them: either do nothing, look completely impotent and lose all "deterrence", or go to war with Iran which is extremely risky and could end in the US' greatest defeat of all time. If we do see them push their navy up into the Persian Gulf it will mean that they have become so desperate that they feel they need to gamble it all on all out war. I don't think that they will do it just as they haven't and won't dare to get directly involved in Ukraine because a bully doesn't pick a fight against anyone who may actually beat or severely hurt them, but you never know what a losing, cornered and increasingly irrational empire may do in its final days.