Are the limits of conversation the limits of Philosophy?

Often in our conversations, we have arrived at an impasse; Reached an aporia if you will, where no further dialogue seems to be possible. Each points the finger at the other says "You are completely missing the point" and silence (often an awkward silence) follows. 

I want to distinguish this feeling from what occurs when people "agree to disagree". When agreeing to disagree you are recognising the legitimacy of the other's position and treating it at par on your own arguments. In some sense it is tolerance. What has happened in our conversations is different from this. It is an inability to convince the other person. Not a recognition of the other's position. Each one is convinced that her proposition is closer to truth, is normatively justified or a more accurate description of the state of affairs. 

The inability of the conversation to move forward, for epistemic reasons I must add, seems indicative of the larger inability of philosophy. Of course, philosophy could go meta and think about what to do when impasse reaches, or why there are impasses, but there is a possibility of an impasse there also. Thus it seems to me that there is no solution to an impasse that can claim to avoid another impasse. 

Another difference from "agreeing to disagree" is that it is possible to say that the debate has been concluded if not resolved. In conversations, however, it is possible that the withdrawal of one person is unilateral. The person might withdraw and say he longer wants to be in the conversation, or even brazenly claim that even if it is irrational he doesn't mind. It seems to me that philosophy has no further tools to coerce a person as long as we are in the realm of beliefs. This seems to be in some sense the limits of philosophy itself.

One might wonder whether it is true not just for philosophy but for all enquiries in general but it occurs to me that in Science atleast, that is not the case. One can point to an empirically observable phenomenon, and aim to settle the debate. I realise that the theory of undetermination would challenge this claim of mine that experiments can settle theoretical disputes but in a large number of cases disputes do get settled in sciences. For instance, if I and you disagree over whether there is a wall outside the window. we can verify the properties of a wall, and which of them should manifest given the particular conditions that exist now and then observe the prediction. What aids the process further is the possibility that the same experiment could be repeated multiple times with changes in contributing factors. Philosophy rarely makes predictions. and in those cases that it has made them, it has not been particularly successful. 

I am not making a wittgenstenian claim here that it is language which sets the limits of philosophy. Philosophical activity has a much smaller playing field. It seems to me, to be an essentially social activity. I am willing to concede that there can be conversations with self and in that sense a Cartesian meditation is philosophy. But some conversation seems essential. Philosophy survives as long as conversations don't end in aporia. The decline in philosophy in contemporary times can be seen to then coincide with the decline in conversations within today's societies.